Digitized  by  tine  Internet  Arciiive 

in  2010  witii  funding  from 

NCSU  Libraries 


http://www.arcliive.org/details/textbookonmeclianOOinte 


A  TEXTBOOK 


ON 


MECHANICAL    AND    ELECTRICAL 
ENGINEERING 


International  Correspondence  Schools 

SCRANTON,  PA. 


GEOMETRICAL  DRAWING 

MECHANICAL   DRAWING 

PRACTICAL  PROJECTION 

DEVELOPMENT  OF  SURFACES 


14965 


SCRANTON 

INTERNATIONAL  TEXTBOOK  COMPANY 

B-3 


Copj^right,  1897,  by  The  Colliery  Engineer  Company. 

Copj-right,  iy02,  by  International  Textbook  Company, 

under  the  title  of  An  Elementary  Treatise  on  Mechanical  Drawing. 


Geometrical  Drawing :  Copjrright,  1893.  1894,  1896, 1897, 1S9S,  1899,  1901.  bv  The  COL- 
LiERV  Engineer  Comp.^xy. 

Mechanical  Drawing:  Cop^-right,  1893.  1S94.  1S98,  by  THE  COLLIERY  ENGINEER 
CoMPANV.  CopjTight.  196-2.  by  Interx.\tional  Textbook  Comp.any.  Entered 
at  Stationers'  Hall,  London. 

Practical  Projection  :     CopjTight,  1899,  by  THE  COLLIERY  ENGINEER  COMPANY. 

Development  of  Surfaces :  Copvright,  1899,  bv  THE  Colliery  Engineer  Company. 

Plate,  Projections— I :  Copyright,  1393, 3894, 1896, 1897, 1898,  bv  THE  COLLIERY  ENGI- 
NEER Co.mpany. 

Plate,  Projections— II  :  Copyright,  1893,  1894,  1896,  1897,  1898,  by  THE  COLLIERY 
Engineer  Company. 

Plate.  Conic  Sections:  Copyright  1893,  1894,  1896,  1897,  1898,  by  THE  COLUERT 
Engineer  company. 

Plate,  Intersections  and  Developments :  Copvright,  1893.  1894,  1897, 1898,  by  THE 
Colliery  Engineer  Company. 

Plate,  Details:  Copyright.  1893,  1896,  1897,  1898,  by  THE  COLLIERY  ENGINEER  COM- 
PANY. 

Plate  Machine  Details :  Copyright,  1893,  1896, 1897,  1898,  bv  THE  Colliery  Engi- 
neer Company. 

Plate,  Flange  Coupling  :  Copyright.  1902,  by  INTERNATIONAL  TEXTBOOK  COM- 
P.\NY.     Entered  at  Stationers'  Hall.  London. 

Plate,  Eccentric  and  Brake  Lever  :  Copyright,  1893,  1895.  1897,  1898,  bv  THE  COL- 
LIERY Engineer  Company. 

Plate.  Timber  Trestle  :  Copyright,  1902,  bj-  International  Textbook  Company. 
Entered  at  Stationers'  Hall,  London. 

Plate,  Steel  Columns  and  Connections:  Cop\Tight,  1902.  bv  International 
Textbook  Company.       Entered  at  Stationers'  Hall.  London. 

Plate,  Turret-Lathe  Tools:  Copvright,  1902.  bv  International  TEXTBOOK  Com- 
pany.   Entered  at  Stationers'  Hall,  London. 

Plate,  Commutator  :  Copyright,  1902,  bv  INTERNATIONAL  TEXTBOOK  COMPANY. 
Entered  at  Stationers'  Hall,  London. 

Plate,  Shaft  Hanger :  Copj-right,  1893, 1895, 1897. 1898.  bv  The  Colliery  Engineer 
Company. 

Plate,  Bench  '\'ise :  Copyright,  1893,  1895,  1897,  1898,  bv  THE  Colliery  Engineer 
Company. 

Plate.  Profiles  of  Gear  Teeth :  Copyright,  1893, 1895, 1897,  1898,  by  The  Colliery 
Engineer  Company. 

Plate.  Spur  Gear  Wheels  :  Copvright,  1893,  1895, 1896,  1897,  1898,  by  The  Colliery 
Engineer  Company. 

Plate.  Bevel  Gears  :  Copvright,  1893, 1895, 1897, 1898,  by  The  Colliery  Engineer 
Company. 

Plate,  Brush  Holder :  Copvright,  1902,  by  INTERNATIONAL  TEXTBOOK  Company. 
Entered  at  Stationers'  Hall.  London. 

Plate,  Compound  Rest :  Copvright.  1902,  by  INTERNATIONAL  TEXTBOOK  COMPANY. 
Entered  at  Stationers'  Hall,  London. 

Plate,  Six  Horsepower  Horizontal  Steam  Engine  :  Copj-right,  1893, 1895,  1897,  1898, 
by  The  Colliery  Engineer  Company. 

Plate,  Proiections— I :    Copvright,  1899,  bv  The  Colliery  Engineer  Company. 

Plate,  Projections— IT  :     Copvrieht,  1899.  bv  THE  COLLIERY   ENGINEER  COMPANY. 

Plate,  Projections— III  :     Copvright.  1899.  bv  The  COLLIERY  ENGINEER  COMPANY. 

Plate.  Sections— I  :    Copvright,  1899.  bv  THE  COLLIERY  ENGINEER  COMPANY. 

Plate,  Sections— II  :     Copvright,  1899,  by  THE  COLLIERY  ENGINEER  COMPANY. 

Plate.  Intersections— I  :     Copvright.  1899,  bv  THE  COLLIERY  ENGINEER  COMPANy, 

Plate.  Intersections— II :    Copvright.  1899.  bv  THE  COLLIERY  ENGINEER  COMPANY. 

Plate,  Developments— I :    Cop\Tight.  1899.  bv  THE  COLLIERY  Engineer  Company. 

Plate.  Developments— II :    Copvright.  1899.  bv  THE  COLLIERY  Engineer  COMPANY. 

Plate,  Developments— III :  Copvright,  1899,  bv  THE  Colliery  Enginfkr  Company. 

Plate,  Developments— IV  :    Copvright,  1899,  bv  THE  COLLIERY  Engineer  COMPANY. 

Plate,  Developments— 'V^ :    Copyright,  1899,  bv  The  COLLIERY  ENGINEER  COMPANY. 


All  rights  reserved. 

<?V 

BURR   PRINTING  HOCSE. 

FRANKFORT  AND  JACOB  STREETS, 

NEW  YORK. 


CONTENTS 


Geometrical  Drawing                                      Section  Page 

Instruments  and  Materials 13  1 

Lettering 13  18 

Plates 13  26 

The  Representation  of  Objects      ...  13  48 

Projections  I 13  55 

Projections  II 13  61 

Conic  Sections 13  65 

Intersections  and  Developments    .     .      .  •  13  67 

Shade  Lines 13  73 

Mechanical  Drawing 

Center  Lines 14  1 

Sections  and  Section  Lining      ....  14  2 

Breaks 14  7 

Hidden  Screw  Threads 14  8 

Repeated  Parts  of  Objects 14  8 

Abbreviations  Used  on  Drawings ...  14  9 

Kinds  of  Working  Drawings      ....  14  10 

Scales 14  12 

Conventional  Representation  of  a  Nut   .  14  15 

Details 14  16 

Machine  Details 14  21 

Flange  Coupling 14  24 

Eccentric  and  Brake  Lever 14  26 

Timber  Trestle 14  27 

iii 


IV 


CONTEXTS 


Mechanical  Drawing — Continued                  Section  P^gt" 

Steel  Columns  and  Connections     ...  14  34 

Turret-Lathe  Tools 14  37 

Commutator 14  43 

Shaft  Hanger 14  45 

Bench  Vise 14  46 

Profiles  of  Gear-Teeth 14  4S 

Definitions  and  Calculations      ....  14  55 

Spur  Gear- Wheels 14  b^ 

Bevel  Gears 14  59 

Brush  Holder 14  63 

Reading  a  Working  Drawing    ....  14  66 

Compound  Rest 14  75 

Tracings 14  77 

Blueprinting 14  78 

SLs-Horsepower   Horizontal   Steam   En- 
gine     14  84 

Practical  Projection 

Orthographic  Projections 

General  Princfples 

Plates    

Drawing  Plate,  Projections  I 
Drawing  Plate,  Projections  II  . 
Drawing  Plate,  Projections  III 
Drawing  Plate,  Sections  I    . 
Drawing  Plate,  Sections  II  . 
Drawing  Plate,  Intersections  I 
Drawing  Plate,  Intersections  II 

Development  of  Sitrfaces 

Definitions 

General  Classification  of  Solids 
Solids  Accurately  Developed     . 
Development  by  Parallel  Lines 
Rules  for  Parallel  Developments 
Drawing  Plate,  Developments  I 
Drawing  Plate,  Developments  II 


15 

1 

15 

3 

15 

14 

15 

15 

15 

47 

15 

63 

15 

70 

15 

81 

15 

87 

15 

93 

16 

1 

16 

3 

16 

5 

16 

13 

16 

24 

16 

•25 

16 

31 

CONTENTS 


Development  of  Surfaces — Continued 
Drawing  Plate,  Developments  III 
Development  by  Radial  Lines  .     . 
Rules  for  Radial  Development 
Drawing  Plate,  Developments  IV 

Triangulation 

Drawing  Plate,  Developments  V  . 
Approximate  Developments      .     . 


Section 

Page 

16 

33 

16 

36 

16 

42 

16 

44 

16 

49 

16 

53 

16 

63 

GEOMETRICAL  DRAWING 


IIS^STRUMENTS  AI^D  MATERIALS 

—  1.  A  drawing  is  a  representation  of  objects  on  a  plane 
surface  by  means  of  lines  or  lines  and  shades.  When  done 
by  the  use  of  free  hand  only,  it  is  called  freehand  draw- 
Injjf  or  sketching ;  when  instruments  are  used,  so  that 
greater  exactness  may  be  obtained,  it  is  called  instru- 
mental, or  mecliaiiieal,  drawing. 

2.  All  the  instruments  and  materials  required  for  the 
courses  in  drawing  are  mentioned  in  the  following  descrip- 
tions: 

The  drawing  board  should  be  made  of  well-seasoned, 
straight-grained  pine,  the  grain  running  lengthwise.  For 
this  Course,  the  student  will  need  a  board  of  the  following 
dimensions:  length  over  all,  22|  inches;  width,  16|  inches. 

The  drawing  board  illustrated  in  Fig.  1  is  the  one  fur- 
nished in  our  students'  drawing  outfits  and  can  be  fully 
recommended  as  possessing  the  qualities  a  good  and  accu- 
rate board  should  have.  It  is  made  of  several  pieces  of  pine 
wood  glued  together  to  the  required  width  of  the  board.  A 
pair  of  hardwood  cleats  is  screwed  to  the  back  of  the  board, 
the  screws  passing  through  the  cleats  in  oblong  slots  with 
iron  bushings,  which  allow  the  screws  to  move  freely  when 
drawn  by  the  contraction  and  expansion  of  the  board. 
Grooves  are  cut  through  half  the  thickness  of  the  board 
over  the  entire  back  side.  These  grooves  take  the  trans- 
verse resistance  out  of  the  wood  and  allow  it  to  be  controlled 

For  notice  of  copyright,  see  page  immediately  following  the  title  page. 
M.  E.      v.— 2 


GEOMETRICAL   DRAWING 


S13 


by  the  cleats,  at  the  same  time  leaving  the  longitudinal 
strength  nearly  unimpaired.  In  order  to  provide  a  per- 
fectly smooth  working  edge  for  the  head  of  the  T  square  to 
slide  against  a  strip  of  hard  wood  is  let  into  the  short  edges 


Fig.  1 


of  the  board,  and  is  sawed  through  in  several  places,  in  order 
to  allow  for  the  contraction  and  expansion  of  the  board.  The 
cleats  also  raise  the  board  from  the  table,  thus  making  it 


pi                -f 

'•]!                           B                                             ^ 

T. 

easier  to  change  the  position  of  the  board.  "When  in  use,  the 
board  is  placed  so  that  one  of  the  short  edges  is  at  the  left 
of  the  draftsman,  as  shown  in  Fig.  2. 

3.  The  T  square  is  used  for  drawing  horizontal  straight 
lines.  The  head  A  is  placed  against  the  left-hand  edge  of 
the  board,  as  shown  in  Fig.  2.  The  upper  edge  C  of  the 
blade  B  is  brought  very  near  to  the  point  through  which  it 
is  desired  to  pass  a  line,  so  that  the  straight  edge  C  of  the 
blade  may  be  used  as  a  guide  fpr  the  pen  or  pencil.  It  is 
evident  that  all  lines  drawn  in  this  manner  will  be  parallel. 


13 


GEOMETRICAL  DRAWING 


Vertical  lines  are  drawn  by  means  of  triangles.  The  tri- 
angles most  generally  used  are  shown  in  Figs.  3  and  4,  each 
of  which  has  one  right  angle.  _  The  triangle  shown  in  Fig.  3 


Fig.  3 


Fig.  4 


has  two  angles  of  45°  each,  and  that  in  Fig.  4  one  of  60°  and 
one  of  30°.      They  are  called  Jk5°  and  60°  triangles,  respect- 


Wq"  '• 


ively.  To  draw  a  vertical 
line,  place  the  T  square  in 
position  to  draw  a  hori- 
zontal line,  and  lay  the  tri- 
angle against  it,  so  as  to 
form  a  right  angle.  Hold 
both  T  square  and  triangle 
lightly  with  the  left  hand, 
so  as  to  keep  them  from 
slipping,  and  draw  the  line 
with  the  pen  or  pencil  held  in  the  right  hand,  and  against 
the  edge  of  the  triangle.  Fig.  5  shows  the  triangles  and 
T  square  in  position. 

4.  For  drawing  parallel  lines  that  are  neither  vertical 
nor  horizontal,  the  simplest  and  best  way,  when  the  lines 
are  near  together,  is  to  place  one  edge  of  a  triangle, 
as  ah.  Fig.  0,  on  the  given  line  c d,  and  lay  the  other  tri- 
angle, as  />.   against  one  of  the  two  edges,  holding  it  fast 


GEOMETRICAL   DRAWING 


§  13 


\rith  the  left  hand;  then  move  the  triangle  A  along  the 
edge  of  B.  The  edge  a  b  will  be  parallel  to  the  line  c  d\ 
and  when  the  edge  a  b  reaches  the  point  g,  through  which  it 
is   desired    to   draw   the   parallel  line,   hold  both  triangles 


t 


Fig.  6 


Stationary  with  the  left  hand  and  draw  the  line  efhy  pass- 
ing the  pencil  along  the  edge  a  b.  Should  the  triangle  A 
extend  too  far  beyond  the  edge  of  the  triangle  B  after  a 
number  of  lines  have  been  drawn,  hold  A  stationary  with 
the  left  hand  and  shift  B  along  the  edge  of  A  with  the 
right  hand  and  then  proceed  as  before. 

5.  A  line  may  be  drawn  at  right  angles  to  another  line 
which  is  neither  vertical  nor  horizontal,  as  illustrated  in 
Fig.  T.  Let  cdhQ  the  given  line  (shown  at  the  left-hand 
side).  Place  one  of  the  shorter  edges,  as  a  b,  of  the  triangle  B 
so  that  it  will  coincide  with  the  line  c  d;  then,  keeping  the 
triangle  in  this  position,  place  the  triangle  A  so  that  its 
long  edge  will  come  against  the  long  edge  of  B.  Now, 
holding  A  securely  in  place  with  the  left  hand,  slide  B  along 
the  edge  of  A  with  the  right  hand,  when  the  lines  //  /,  vi  Ji, 
etc.  may  be  drawn  perpendicular  to  cd  along  the  edge  bf 
of  the  triangle  B.  The  dotted  lines  show  the  position  of  the 
triangle  B  when  moved  along  the  edge  of  A. 

6.  The  right-hand  portion  of  Fig.  7  shows  another 
method  of  accomplishing  the  same  result,  and  illustrates 


§13 


GEOMETRICAL  DRAWING 


how  the  triangles  may  be  used  for  drawing  a  rectangular 
figure,  when  the  sides  of  the  figure  make  an  angle  with  the 
T  square  such  that  the  latter  cannot  be  used. 

Let  the  side  cdoi  the  figure  be  given.  Place  the  long  side 
of  the  triangle  B  so  as  to  coincide  with  the  line  c  d,  and 
bring  the  triangle  A  into  position  against  the  lower  side  of  B, 
as  shown.  Now,  holding  the  triangle  A  in  place  with  the 
left  hand,  revolve  B  so  that  its  other  short  edge  will  rest 
against  the  long  edge  oi  A,  as  shown  in  the  dotted  position 
at  i?'.     The  parallel  line's  ce  and  «?/ may  now  be  drawn 


Fig.  7 
through  the  points  c  and  d  by  sliding  the  triangle  B  on  the 
triangle  A,  as  described  in  connection  with  Fig.  6.  Meas- 
ure off  the  required  width  of  the  figure  on  the  line  c  e, 
reverse  the  triangle  B  again  to  its  original  position,  still 
holding  the  triangle  ^  in  a  fixed  position  with  the  left  hand, 
and  slide  B  upon  A  until  the  long  edge  of  .5  passes  through  ^.' 
Draw  the  line  ef  through  the  point  e,  and  r/will  be  par- 
allel to  cd.  The  student  should  practice  with  his  triangles 
before  beginning  drawing. 

7.  The  compasses,  next  to  the  T  square  and  triangles, 
are  used  more  than  any  other  instrument.  A  pencil  and 
pen  point  are  provided,  as  shown  in  Fig.  8,  either  of  which 


GEOMETRICAL   DRAWING 


13 


may  be  inserted  into  a  socket  in  one  leg  of  the  instrument, 
for  the  drawing  of  circles  in  pencil  or  ink.  The  other  leg 
is  fitted  with  a  needle  point,  which  acts  as  the  center  about 
which  the  circle  is  drawn.  In  all  good  instruments,  the 
needle  point  itself  is  a  separate  piece  of  round  steel  wire, 
held  in  place  in  a  socket  provided  at  the  end  of  the  leg. 
The  wire  should  have  a  square  shoulder  at  its  lower  end, 
below  which  a  fine,  needle-like  point  projects.  The  length- 
ening bar,  also  shown  in  the  figure,  is  used  to  extend  the 
leg  carrying  the  pen  and  pencil  points  when  circles  of  large 
radii  are  to  be  drawn. 

The  joint  at  the  top  of  the  compasses  should  hold  the  legs 
firmly  in  any  position,  and  at  the  same  time  should  permit 

their  being  opened  or  closed 
with  one  hand.  The  joint  may 
be  tightened  or  loosened  by 
means  of  a  screwdriver  or 
wrench,  which  accompanies 
the  compasses. 

It  will  be  noticed  in  Fig.  8 
that  each  leg  of  the  compasses 
is  jointed ;  this  is  done  so  that 
the  compass  points  may  always 
be  kept  perpendicular  to  the 
paper  when  drawing  circles,  as 
in  Fig.  11. 

The  style  of  compasses  shown 
in  Fig.  8  have  what  is  called  a 
tongue  joint,  in  which  the  head 
of  one  leg  has  a  tongue,  gener- 
ally of  steel,  which  moves  bC' 
tween  two  lugs  on  the  other 
leg.  Another  common  style  of 
joint  is  the  pivot  joint ,  in  which 
the  head  of  each  leg  is  shaped 
like  a  disk  and  the  two  disks 
are  held  together  in  a  fork-shaped  brace  either  by  means  of 
two  pivot  screws  or  by  one  screw  penetrating  both  disks. 


Fig.  8 


§13 


GEOMETRICAL  DRAWING 


The  brace  that  forms  a  part  of  this  joint  is  generally  pro- 
vided with  a  handle,  as  the  shape  of  the  joint  makes  it  rather 


Fig.  9 

awkward  to  hold  the  compasses  by  the  head,  as  is  usual  with 
instruments  provided  with  tongue  joints.  In  Fig.  9  is 
shown  a  common  style  of  pivot  joint. 

8.  The  following  suggestions  for  handling  the  compasses 
should  be  carefully  observed  by  those  who  are  beginning  the 
subject  of  mechanical  drawing.  Any  draftsman  who  handles 
his  instruments  awkwardly  will  create  a  bad  impression,  no 
matter  how  good  a  workman  he  may  be.     The  tendency  of 


Fig.  10 

all  beginners  is  to  use  both  hands  for  operating  the  com- 
passes. This  is  to  be  avoided.  The  student  should  learn 
at  the  start  to  open  and  close  them  with  one  hand,  holding 
them  as  shown  in  Fig.  10,  with  the  needle-point  leg  resting 
between  the  thumb  and  fourth  finger,  and  the  other  leg 
between  the  middle  and  forefinger.     When  drawing  circles, 


GEOMETRICAL   DRAWING 


§  13 


hold  the  compasses  lightly  at  the  top  between  the  thumb  and 
forefinger,  or  thumb,  forefinger,  and  middle  finger,  as  in 
Fig.  11.  Another  case  where  both  hands  should  not  be  used 
is  in  locating  the  needle  point  at  a  point  on  the  drawing  about 
which  the  circle  is  to  be  drawn,  unless  the  left  hand  is  used 
merely  to  steady  the  needle  point.  Hold  the  compasses  as 
shown  in  Fig.  10,  and  incline  them  until  the  under  side  of  the 


Fig.  11 


hand  rests  upon  the  paper.  This  will  steady  the  hand  so  that 
the  needle  point  can  be  brought  to  exactly  the  right  place 
on  the  drawing.  Having  placed  the  needle  at  the  desired 
point,  and  with  it  still  resting  on  the  paper,  the  pen  or  pen- 
cil point  may  be  moved  out  or  in  to  any  desired  radius, 
as  indicated  in  Fig.  10.  When  the  lengthening  bar  is  used, 
both  hands  must  be  employed. 

9.     The  compasses  must  be  handled  in  such  a  manner  that 
the  needle  point  will  not  dig  large  holes  in  the  paper.     Keep 


§  13  GEOMETRICAL  DRAWING  9 

the  needle  point  adjusted  so  that  it  will  he  perpendicular  to 
the  paper,  when  drawing  circles,  and  do  not  hear  upon  it. 
A  slight  pressure  will  be  necessary  on  the  pen  or  pencil 
point,  but  not  on  the  needle  point. 

10.  The  dividers,  shown  in  Figs.  9  and  12,  are  used 
for  laying  off  distances  upon  a  drawing,  or  for  dividing 
straight  lines  or  circles  into  parts.  The  points  of  the 
dividers  should  be  very  sharp,  so  that  they  will  not  punch 
holes  in  the  paper  larger  than  is  absolutely  necessary  to  be 
seen.  Compasses  are  sometimes  furnished  with  two  steel 
divider  points,  besides  the  pen  and  pencil  points,  so  that  the 
instrument  may  be  used  either  as  compasses  or  dividers. 
This    is   the    kind  illustrated  in   Fig.  12.     When  using  the 


Fig.  12 


dividers  to  space  a  line  or  circle  into  a  number  of  equal  parts, 
hold  them  at  the  top  between  the  thumb  and  forefinger,  as 
when  using  the  compasses,  and  step  oflf  the  spaces,  turning 
the  instrument  alternately  to  the  right  and  left.  If  the  line 
or  circle  does  not  space  exactly,  vary  the  distance  between 
the  divider  points  and  try  again;  so  continue  until  it  is 
spaced  equally.  When  spacing  in  this  manner,  great  care 
must  be  exercised  not  to  press  the  divider  points  into  the 
paper;  for,  if  the  points  enter  the  paper,  the  spacing  can 
never  be  accurately  done.  The  student  should  satisfy  him- 
self of  the  truth  of  this  statement  by  actual  trial. 

11.  The  bow-pencil  and  bow-i)en,  shown  in  Fig.  13, 
are  convenient  for  describing  small  circles.  The  two  points 
of  the  instruments  must  be  adjusted  to  the  same  length; 
otherwise,  very  small  circles  cannot  be  drawn.  To  open  or 
close  either  of   these  instruments,  support  it  in  a  vertical 


10 


GEOMETRICAL   DRAWING 


13 


Fig.  13 


position  by  resting  the  needle  point  on  the  paper  and  bear- 
ing slightly  on  the  top  of  it  with  the  forefinger  of  one  hand, 

and  turn  the  adjusting  nut 
with  the  thumb  and  middle 
finger  of  the  same  hand. 

13,  Dra^^ing  Paper 
and  Pencils. — The  draw- 
ing paper  recommended  for 
this  series  of  lessons  is  T.  S. 
Co.  's  cold-pressed  demy,  the 
size  of  which  is  15"  X  20". 
It  takes  ink  well  and  with- 
stands considerable  erasing. 
The  paper  is  secured  to  the 
drawing  board  by  means 
of  tliiimb tacks.  Four  are 
usually    sufficient  —  one    at 

each  corner  of  the  sheet  (see  Fig.  7).     Place  a  piece  of  paper 

on  the  drawing  board,  and  press  a  thumbtack  through  one 

of  the  corners  about  :^  or  f  of  an  inch 

from  each  edge.     Place  the  T  square  in 

position  for  drawing  a  horizontal  line, 

as  before  explained,  and  straighten  the 

paper    so   that    its   upper  edge  will   be 

parallel    to    the   edge    of    the   T-square 

blade.  Pull  the  corner  diagonally  op- 
posite   that    in    which    the  "thumbtack 

was  placed,  so  as  to  stretch  the  paper 

slightly,  and  push  in  another  thumb- 
tack.     Do  the  same  with  the  remaining 

two  corners.     For  drawing  in  pencil,  an 

HHHH  pencil  of  any  reputable  make 

should  be  used      The  pencil  should  be 

sharpened  as  shown  at  A,  Fig.  14.     Cut 

the  wood  away  so  as  to  leave  about  ^  or 

f  of  an  inch  of  the  lead  projecting;  then 

sharpen  it  flat  by  rubbing  it  against  a  fine  file  or  a  piece  of 


Fig. 14 


GEOMETRICAL  DRAWING 


11 


fine  emery  cloth  or  sandpaper  that  has  been  fastened  to  a 
flat  stick.  Grind  it  to  a  sharp  edge  like  a  knife  blade,  and 
round  the  corners  very  slightly,  as  shown  in  the  figure.  If 
sharpened  to  a  round  point,  as  shown  at  B,  the  point  will 
wear  away  very  quickly  and  make  broad  lines;  when  so 
sharpened  it  is  difficult  to  draw  a  line  exactly  through  a 
point.  The  lead  for  the  compasses  should  be  sharpened  in 
the  same  manner  as  the  pencil,  but  should  have  its  width 
narrower.  Be  sure  that  the  compass  lead  is  so  secured  that 
when  circles  are  struck  in  either  direction,  but  one  line  ivill 
be  draxvn  with  the  same  radius  and  center. 

13.  Inking. — For  drawing  ink  lines  other  than  arcs  of 
circles,  the  ruling  pen  (or  right-line  pen,  as  it  is  sometimes 
called)  is  used.  It  should  be  held  as  nearly  perpendicular 
to  the    board   as  possible,   with  the  hand    in   the   position 


Fig.  15 


shown  in  Figs.  15  and  16,  bearing  lightly  against  the  T  square 
or  triangle,  along  the  edge  of  which  the  line  is  drawn. 
After  a  little  practice,  this  position  will  become  natural,  and 
no  difficulty  will  be  experienced. 


12 


GEOMETRICAL   DRAWING 


13 


1-4.  The  beginner  will  find  that  it  is  not  always  easy  to 
make  smooth  lines.  If  the  pen  is  held  so  that  only  one  blade 
bears  on  the  paper  when  drawing,  the  line  will  almost  inva- 
riably be  ragged  on  the  edge  where  the  blade  does  not  bear. 
When  held  at  right  angles  to  the  paper,  as  in  Fig.  16,  how- 
ever, both  blades  will  rest  on  the  paper,  and  if  the  pen  is  in 
good  condition,  smooth  lines  will  result.  The  pen  must  not 
be  pressed  against  the  edge  of  the  T  square  or  triangle,  as 
the  blades  will  then  close  together,  making  the  line  uneven. 
The  edge  should  serve  as  a  guide  simply. 


Fig.   16 


In  drawing  circles  with  the  compass  pen.  the  same  care 
should  be  taken  to  keep  the  blades  perpendicular  to  the 
paper  by  means  of  the  adjustment  at  the  joint.  In  both 
the  ruling  pen  and  compass  pen.  the  width  of  the  lines  can 
be  altered  by  means  of  the  screw  which  holds  the  blades 
together.  The  handles  of  most  ruling  pens  can  be  unscrewed, 
and  are  provided  with  a  needle  point  intended  for  use  when 
copying  maps  by  pricking  through  the  original  and  the 
underlying  paper,  thus  locating  a  series  of  points  through 
which  the  outline  may  be  drawn. 


§  13  GEOMETRICAL   DRAWING  13 

15.  Di-a^wing  Ink. — The  ink  we  recommend  for  the 
work  in  this  Course  is  the  T.  S.  Co.'s  superior  waterproof 
liquid  India  ink.  A  quill  is  attached  to  the  cork  of  every 
bottle  of  this  ink,  by  means  of  which  the  pen  may  be  filled. 
Dip  the  quill  into  the  ink  and  then  pass  the  end  of  it 
between  the  blades  of  the  drawing  pen.  Do  not  put  too 
much  ink  in  the  pen,  not  more  than  enough  to  fill  it  for  a 
quarter  of  an  inch  along  the  blades,  otherwise  the  ink  is 
liable  to  drop.  Many  draftsmen  prefer  to  use  stick  India 
ink;  and  for  some  purposes  this  is  to  be  preferred  to  the 
prepared  liquid  ink  recommended  above.  In  case  the  stick 
ink  is  bought,  put  enough  water  in  a  shallow  dish  (a  com- 
mon individual  butter  plate  will  do)  to  make  enough  ink  for 
the  drawing,  then  place  one  end  of  the  stick  in  the  water, 
and  grind  by  giving  the  stick  a  circular  motion.  Do  not 
bear  hard  upon  the  stick.  Test  the  ink  occasionally  to  see 
if  it  is  black.  Draw  a  fine  line  with  the  pen  and  hold  the 
paper  in  a  strong  light.  If  it  shows  brown  (or  gray),  grind 
a  while  longer,  and  test  again.  Keep  grinding  until  a  fine 
line  shows  black,  which  will  usually  take  from  fifteen  min- 
utes to  half  an  hour,  depending  on  the  quantity  of  water 
used.  The  ink  should  always  be  kept  well  covered  with  a 
flat  plate  of  some  kind,  to  keep  out  the  dust  and  prevent 
evaporation.  The  drawing  pen  may  be  filled  by  dipping  an 
ordinary  writing  pen  into  the  ink  and  drawing  it  through 
the  blades,  as  previously  described  when  using  the  quill.  If 
liquid  ink  is  used,  all  the  lines  on  all  the  drawings  will  be  of 
the  same  color,  and  no  time  will  be  lost  in  grinding.  If 
stick  ink  is  used,  it  is  poor  economy  to  buy  a  cheap  stick. 
A  small  stick  of  the  best  quality,  costing,  say,  a  dollar,  will 
last  as  long,  perhaps,  as  five  dollars'  worth  of  liquid  ink. 
The  only  reason  for  using  liquid  ink  is  that  all  lines  are  then 
sure  to  be  of  equal  blackness  and  time  is  saved  in  grinding, 

India  ink  will  dry  quickly  on  the  drawing,  which  is  desir- 
able, but  it  also  causes  trouble  by  drying  between  the  blades 
and  refusing  to  flow,  especially  when  drawing  fine  lines. 
The  only  remedy  is  to  wipe  out  the  pen  frequently  with  a  cloth. 
Do  not  lay  the  pen  down  for  any  great  length  of  time  when 


14  GEOMETRICAL   DRAWING  §  13 

it  contains  ink;  wipe  it  out  first.  The  ink  may  sometimes  be 
started  by  moistening  the  end  of  the  finger  and  touching  it 
to  the  point,  or  by  drawing  a  slip  of  paper  between  the  ends 
of  the  blade.     Always  keep  the  bottle  corked. 

16.  To  Sliai-pen  tlie  I>i*a^ving  Pen. — "When  the  ruling, 
or  compass,  pen  becomes  badly  worn,  it  must  be  sharp- 
ened. For  this  purpose  a  fine  oilstone  should  be  used.  If 
an  oilstone  is  to  be  purchased,  a  small,  flat,  close-grained 
stone  should  be  obtained,  those  having  a  triangular  section 
being  preferable,  as  the  narrow  edge  can  be  used  on  the 
inside  of  the  blades  in  case  the  latter  are  not  made  to  swing 
apart  so  as  to  permit  the  use  of  a  thicker  edge. 

The  first  step  in  sharpening  is  to  screw  the  blades  together, 
and,  holding  the  pen  perpendicular  to  the  oilstone,  to  draw 
it  back  and  forth  over  the  stone,  changing  the  slope  of  the 
pen  from  downwards  and  to  the  right  to  downwards  and  to 
the  left  for  each  movement  of  the  pen  to  the  right  and  left. 
The  object  of  this  is  to  bring  the  blades  to  exactly  the 
same  length  and  shape,  and  to  round  them  nicely  at  the 
point. 

This  process,  of  course,  makes  the  edges  even  duller  than 
before.  To  sharpen,  separate  the  points  by  means  of  the 
screw,  and  rub  one  of  the  blades  to  and  from  the  operator 
in  a  straight  line,  giving  the  pen  a  slight  twisting  motion 
at  the  same  time,  and  holding  it  at  an  angle  of  about  15°  with 
the  face  of  the  stone.  Repeat  the  process  for  the  other 
blade.  To  be  in  good  condition,  the  edges  should  be  fairly 
sharp  and  smooth,  but  not  sharp  enough  to  cut  the  paper. 
All  the  sharpening  must  be  done  on  the  outside  of  the  blades. 
The  inside  of  the  blades  should  be  rubbed  on  the  stone  only 
enough  to  remove  any  burr  that  may  have  been  formed. 
Anything  more  than  this  will  be  likely  to  injure  the  pen. 
The  whole  operation  must  be  done  very  carefully,  bearing 
on  lightly,  as  it  is  easy  to  spoil  a  pen  in  the  process 
Examine  the  points  frequently,  and  keep  at  work  until  the 
pen  will  draw  hoth  fine  lines  and  smooth  heavy  lines.  Many 
draftsmen  prefer  to  send  the  pens  to  be  sharoened  to  the 


§13 


GEOMETRICAL   DRAWING 


15 


dealer  who  sold  them,  and  who  is  generally  willing  to  do 
such  sharpening  at  a  trifling  cost. 


17.  Irregular  Curves. — Curves  other  than  arcs  of  cir- 
cles are  drawn  with  the  pencil  or  ruling  pen  by  means  of 
curved  or  irregular-shaped  rulers,  called  irrejyular  curves 
(see  Fig.  17).  A  series  of  points  is  first  determined  through 
which  the  curved  line  is  to  pass,  'i'he  line  is  then  drawn 
through  these  points  by  using  such  parts 
of  the  irregular  curve  as  will  pass  through 
several  of  the  points  at  once,  the  curve  be- 
ing shifted  from  time  to  time  as  required. 

It  is  usually  ditBcult  to  draw  a  smooth, 
continuous  curve.  The  tendency  is  to 
make  it  curve  out  too  much  between  the 
points,  thus  giving  it  a  wavy  appearance, 
or  else  to  cause  it  to  change  its  direction 
abruptly  where  the  different  lines  join, 
making  angles  at  these  points.  These 
defects  may  largely  be  avoided  by  always 
fitting  the  curve  to  at  least  three  points, 
and,  when  moving  it  to  a  new  position, 
by  setting  it  so  that  it  will  coincide  with 
part  of  the  line  already  drawn.  It  will  be 
found  to  be  a  great  help  if  the  line  be 
first  sketched  in  freehand,  in  pencil.  It 
can  then  be  penciled  over  neatly,  or  inked, 
without  much  difficulty,  with  the  aid  of  the  irregular  curve, 
since  the  original  pencil  line  will  show  the  general  direction 
in  which  the  curve  should  be  drawn.  Whenever  the  given 
points  are  far  apart,  or  fall  in  such  positions  that  the  irreg- 
ular curve  cannot  always  be  made  to  pass  through  three  of 
them,  the  line  must  invariably  be  sketched  in  at  first. 

As  an  example,  let  it  be  required  to  draw  a  curved  line 
through  the  points  a,  b,  c,  d,  etc..  Fig.  IS.  As  just  stated,  a 
part  of  the  irregular  curve  must  be  used  which  will  pass 
through  at  least  three  points.  With  the  curve  set  in  the  first 
position  ./,  its  edge  is  found   to  coincide  with   four  points 


Fig.  17 


16 


GEOMETRICAL   DRAWING 


tf,  b,  c,  and  </.  The  line  may  then  be  drawn  from  a  around  to 
d,  or,  better,  to  a  point  between  c  and  d,  since,  by  not  con- 
tinuing it  quite  to  d,  there  is  less  liability  of  there  being  an 
angle  where  the  next  section  joins  on.  For  the  next  section 
of  the  line,  the  curve  should  be  adjusted  so  as  to  coincide 
with  a  part  of  the  section  already  drawn ;  that  is,  instead  of 
adjusting  it  to  points  ^,  i',/",  etc.,  it  should  be  placed  so  as  to 


Fig.  18 


pass  through  the  point  c,  the  part  from  c  to  //being  coincident 
with  the  corresponding  part  of  the  first  line  drawn.  The 
irregular  curve  is  shown  dotted  in  this  position  at  B.  Its  edge 
passes  through  four  points  <:,  d,  e,  andy",  and  the  line  should 
be  made  to  stop  midway  between  the  last  two,  as  before. 
Now,  it  will  be  noticed  that  the  points  /  and  ^  are  so  situ- 
ated that  the  remainder  of  the  line  must  curve  up,  instead  of 
down,  as  heretofore,  the  change  in  curvature  occurring  at  a 


13 


GEOMETRICAL   DRAWING 


17 


point  between  e  and  /.  In  this  case,  therefore,  it  is  not 
necessary  for  the  curve  to  extend  back  to  e,  through  which 
point  the  line  has  already  been  drawn,  but  it  may  be  placed 
in  position  C  with  its  edge  just  tangent  to  the  line  at  the  point 
where  the  curvature  changes. 

It  is  to  be  noticed  that  in  inking  with  the  irregular  curve, 
the  blades  of  the  pen  must  be  kept  tangent  to  its  edge  (i.  e., 
the  inside  flat  surface  of  the  blades  must  have  the  same 
direction  as  the  curve  at  the  point  where  the  pen  touches  the 
paper),  which  requires  that  the  direction  of  the  pen  be  con- 
tinually changed. 

18.  The  scale  is  used  for  obtaining  measurements  for 
drawings.  The  most  convenient  forms  are  the  usual  flat  and 
triangular  boxwood  scales,  having  beveled  edges,  each  of 
which  is  graduated  for  a  distance  of  12  inches.  The  beveled 
edges  serve  to  bring  the  lines  of  division  close  to  the  paper 
when  the  scale  is  lying  flat,  so  that  the  drawing  may  be 
accurately  measured,  or  distances  laid  off  correctly.  The 
use  of  the  graduations  on  scales  will  be  explained  when  it  is 
necessary  to  use  the  scale. 

19.  A  protractor  is  shown  in  Fig.  19.  The  outer 
edge  is  a  semicircle,  with  center  at   O,  and  is  divided  into 


360  parts.      Each  division  is  one-half  of  one  degree,  and,  for 
convenience,  the  degrees  are  numbered  from  0°  to  180"  from 

M.  E.    V.-3 


18 


GEOMETRICAL   DRAWING 


§1' 


both  A  and  B.  The  protractor  is  used  for  la)'ing  off  or 
measuring  angles.  Protractors  are  often  made  of  metal, 
in  which  case  the  central  part  is  cut  away  to  make  the 
drawing  under  it  visible.  When  using  the  protractor,  it 
must  be  placed  so  that  the  line  OB.  Fig.  19,  will  coin- 
cide with  the  line  forming  one  side  of  the  angle  to  be  laid 
off  or  measured,  and  the  center  O  must  be  at  the  vertex  of 
the  angle. 


Fig.  ej 

For  example,  let  it  be  required  to  draw  a  line  through 
the  point  C,  making  an  angle  of  54°  with  the  line  £  F, 
Fig.  '20.  Place  the  protractor  upon  the  line  £F,  as  just 
described,  with  the  center  O  upon  the  point  C.  With  a 
sharp-pointed  pencil,  make  a  mark  on  the  paper  at  the 
54'  division,  as  indicated  at  D.  A  line  drawn  through  C 
and  D  will  then  make  an  angle  of  54°  with  £  F.  Greater 
exactness  will  be  secured"  if  the  line  £F  be  extended  to 
the  left,  so  that  both  zero  marks  (A  and  B,  Fig.  19)  can 
be  placed  on  the  line.  This  should  always  be  done  when 
possible. 


liETTERIXG 

20.  In  mechanical  drawing,  all  headings,  explanatory 
matter,  and  dimensions  should  be  neatly  printed  on  the 
drawing.     Ordinary  script  writing  is  not  permissible. 


§  13  GEOMETRICAL   DRAWING  19 

It  is  usually  difficult  for  beginners  to  letter  well,  and 
unless  the  student  is  skilful  at  it.  he  should  devote  some 
time  to  practicing  lettering  before  commencing  the  drawing. 
In  correcting  the  plates,  the  lettering  will  be  considered  as 
well  as  the  drawing.  Many  students  think  that  it  is  only 
necessary  to  exercise  special  care  when  drawing  the  views 
on  a  plate,  and  that  it  is  not  necessary  to  take  particular 
pains  in  lettering.  This,  however,  is  not  the  case,  for,  no 
matter  how  well  the  views  may  be  drawn,  if  the  lettering  is 
poorly  done,  the  finished  drawing  will  not  have  a  neat  appear- 
ance. In  fact,  generally  speaking,  more  time  is  required  to 
make  well-executed  letters  than  to  make  well-executed  draw- 
ings of  objects.  We  earnestly  request  the  student  to  prac- 
tice lettering,  and  not  to  think  that  that  part  of  the  work  is 
of  no  importance.  The  student  should  not  be  too  hasty  in 
doing  the  lettering.  It  takes  an  experienced  draftsman  con- 
siderable time  to  do  good  lettering,  and  no  draftsman  can 
perform  this  work  as  quickly  as  he  can  ordinary  writing; 
therefore,  no  beginner  should  attempt  to  do  what  experi- 
enced draftsmen  cannot  do.  In  order  to  letter  well,  the 
work  must  be  done  slowly.  Very  frequently  more  time 
is  spent  in  lettering  a  drawing  than  in  inking  in  the 
objects  represented.  Instructions  will  be  given  in  two 
styles  of  freehand  lettering,  both  extensively  used  in  Ameri- 
can drafting  rooms. 

With  the  exception  of  the  large  headings  or  titles  of  the 
plates,  the  style  and  size  of  all  lettering  used  on  the  original 
drawing  plates  of  this  Course  are  shown  in  Fig.  21.      This 

ABCDEFGJ-CIJJfLJyrNOPQRSTirVWXYZ 
abcde/^hijJtlTTinopqr stuvLosc y <:  & 
/ S3456?'690  /e3456  789o e-6i"dtcz.Cast  Iron 

Fig.  21 

Style,  although  a  little  more  elaborate  and  difficult  in  execu- 
tion, was  selected  on  account  of  its  greater  neatness  and 
legibleness.  The  two  styles  are  very  similar  in  the  forma- 
tion of  the  letters,  and  although  the  student  is  advised  to 


20 


GEOMETRICAL  DRAWING 


§13 


select  and  use  only  one  of  the  two  on  his  drawings  in  this 
Course,  he  will  find,  after  having  mastered  one  of  the  styles, 
little  difficulty  in  practicing  the  other. 

When  lettering,  a  Gillott's  No.  303  pen  should  be  used. 
The  height  of  the  capital  letters  should  be  -jV',  and  of 
the  small  letters  two-thirds  of  this,  or  ^".  This  applies 
to  both  styles  of  freehand  lettering.  Do  not  make  them 
larger  than  this. 

21.  Before  beginning  to  letter,    horizontal  guide   lines 
should  be  drawn  with  the  T  square,  to  serve  as  a  guide  for 
the  tops  and  bottoms  of  the  letters  (see  Fig.  22).     The  out- 
side lines  should  be  3^"  apart 

2M^uzrn^^  M&i^^^suca^-    ^^^  ^j^^  capitals,  and  the  two 

^'^-  ^  lower  lines  ^"  apart  for  the 

small  letters.  The  letters  should  be  made  to  extend  fully 
up  to  the  top,  and  down  to  the  bottom,  guide  lines.  They 
must  not  fall  short  of  the  guide  lines,  nor  extend  beyond 
them.  Failure  to  observe  this  point  will  cause  the  lettering 
to  look  ragged,  as  in  the  second  word  in  Fig.  22. 

22.  It  is  very  important    that  all  the  letters  have  the 
same  inclination.     For  example,  by  referring  to  Fig.  23  {a), 
it  will  be  seen  that  the  backs  of    .„.___.._._..^ .,_.....__. ., 
letters  like  B,  E,  /,  g,  d,  t,  t,  etc.         -^    •-      '    '  ^r 
are  parallel  and    slant    the    same                  fig.  23  U) 

way.  This  is  also  true  of  both  sides  of  letters  like  H,  M, 
iiy  u,  h,  J',  etc.     To  aid  in  keeping  the  slant  uniform,  draw 

parallel  slanting  lines 
across  the  guide  lines 
with  the  60°  triangle, 
as  in  Fig.  23  {b),  and, 
in  lettering,  make  the 
backs  or  sides  of  the 
letters  parallel  with 
these  lines. 

23.  A  few  points 
regarding  the  construction  of  the  letters  are  illustrated  in 
Fig.  24,  in  which  the  letters  are  shown  upon  an  enlarged 


f — 

mm#\ 

/ 

-r-,'-'f'--T-jr-<r- 

1'  .  '                                                                                         0 

Fig.  23  ib) 

§  13  GEOMETRICAL  DRAWING  21 

scale.  The  capital  letters  A,  V,  V,  31,  and  W  must  be 
printed  so  that  their  general  inclination  will  be  the  same 
as  for  the  other  letters.  To  print  the  A,  draw  the  center 
line  ad,  having  the  common  slant;  from  a  draw  the  sides  ac 
and  a  b,  so  that  points  c  and  /;  will  each  be  -^^'  distant  from 
point  d.  The  side  a  b  will  be  nearly  perpendicular  to  the 
guide  lines.  The  V  is  like  an  inverted  A,  and  is  drawn  in 
the  same  way,  the  line  b  dh^\xi<g  nearly  perpendicular. 

To  make  the  Y,  draw  the  center  line  a  d,  having  the  com- 
mon slant,  which  gives  the  slant  for  the  base  of  the  letter. 
The  upper  part  of  the  Y  begins  a  little  below  its  center,  and 
is  similar  to  the  ]\  though  somewhat  narrower,  as  the  letter 
should  be  only  -^'  wide  at  the  top.  Points  b  and  c  should 
be  at  equal  distances  from  point  a. 

The  two  sides  be  and  ^y  of  the  M  are  parallel,  and  have 
the  common  slant.  The  M  is  made  as  broad  as  it  is  high, 
or  ^y.       Having  drawn 

point  d,  midwky  between  ,,,-45v^y(^----.i^--^ 

the  points  c  and  /,  and  ]^-uAii   nnyv    .'^hjBtEA 

connect  it  with  points  b 

andr.     The  lines  (^ ^/ and     -uu,ull^^gg    rr^y^dp 

ed    should     be     slightly 

*"       ^  Fig.  34 

curved,  as  shown. 

In  the  J/ 'the  two  outside  lines  are  not  parallel,  as  in   the 

M,  but  are  farther  apart   at  the   top  than  at   the  bottom. 

Draw  the  line  ad,  having  the  common  slant.      Mark  points  b 

and  c,  which  are  exactly  -jV'  from  the  point  a.      From  b  and  c, 

draw  lines  ^</and  c d.     The  other  half  of  the  W\^  like  the 

first  part,  cf  being  parallel  to  b d 2i\\A  ^/parallel  to  c d.      It 

will  be  seen  that  the  W'l's,  composed  of  two  narrow  Ps,  each 

^^"  wide,  the  width  of  the  whole  letter  being  |". 

34.  Capital  letters  like  P,  R,  B,  L,  E,  etc.  should  be 
printed  so  that  their  top  and  bottom  lines  will  be  exactly 
horizontal.  This  is  illustrated  in  the  two  examples  of  the 
word  problem  in  Fig.  24.  In  the  first  example,  it  will  be 
noticed  that  the  tops  of  the  Pand  A',  the  bottom  of   the  L, 


22  GEOMETRICAL   DRAWING  §  13 

and  the  tops  and  bottoms  of  the  B  and  E^  all  run  in  the  same 
direction  as  the  guide  lines,  and  coincide  with  them.  In  the 
second  example,  these  lines  are  not  horizontal,  which  makes 
the  word  look  very  uneven.  It  is  also  to  be  noticed  that 
these  lines  extend  beyond  the  upright  lines  in  the  first 
word,  and  that  cross-lines  are  used  on  the  bottom  of  the 
P  and  R,  on  the  top  of  the  Z,  and  on  the  M.  In  the 
second  word,  these  lines  are  omitted  at  the  points  indicated 
by  the  arrows.  These  features  are  found  on  most  of  the 
other  capitals. 

The  small  letters  n,  ii,  h,  I,  i,  etc.  should  have  sharp  cor- 
ners at  the  points  indicated  by  the  arrows  in  Fig.  24.  They 
look  much  better  that  way,  and  are  less  difficult  to  make, 
than  when  they  have  round  corners.  Following  these  letters 
are  five  groups  of  letters  containing  n,  n,  /,  g,  and  r.  The 
first  letter  of  each  group  is  printed  correctly,  while  the  letters 
following  show  ways  in  which  they  should  not  be  printed. 
In  the  case  of  the^,  point  2  should  fall  in  a  slanting  direction 
under  point  1,  the  slant  being  the  same  as  a  d  oi  the  pre- 
ceding letters.  The  difference  between  d  and  b  and  the  con- 
struction of  the  s  are  also  shown  in  the  same  figure.  The  b 
should  be  made  rounding  at  the  point  indicated.  As  a 
guide  in  making  the  s,  draw  the  two  lines  ab  and  c  d, 
having  the  common  slant.  The  s  should  now  be  drawn 
so  that  it  will  touch  these  lines  at  points  i,  5,  and  .^,  but 
not  at  point  2.  It  will  be  an  additional  help  if  the  line  ex 
is  also  drawn  as  a  guide  for  the  middle  portion  of  the  s\ 
but  care  should  be  taken  not  to  have  it  slant  more  than 
shown  in  the  copy. 

The  letters  a^  o,  b,  g,  etc.  should  be  full  and  round ;  do  not 
cramp  them.  It  will  be  necessary  to  follow  the  copy  closely 
until  familiar  with  it.  Notice  that  the  figures  are  not  made 
as  in  writing,  particularly  the  6,  4,  8,  and  9  (see  Fig.  21). 
Try  to  space  the  letters  evenly.  Letter  in  pencil  first,  and, 
if  not  right,  erase  and  try  again. 

35.*  Another  style  of  freehand  lettering  is  shown  in 
Fig.  25.     This  style  is  extensively  used  for  the  lettering  of 


§  13  GEOMETRICAL   DRAWING  23 

working  drawings.     It  is  more  easily  and  rapidly  made  than 
the  style  previously  described,  and  although  not  productive 

A  BCDEFGH/JKL  M/VOPQRS  TUVVV,KyZ 
/23456769/0  /234567890  2-64  a'/'a.  Casf/ro/7. 

Fig.  io 

of  as  high  degree  of  neatness  in  appearance  will  be  found 
very  useful  and  acceptable  for  general  office  work. 

A  comparison  between  the  two  systems  will  disclose  a  great 
similarity  in  the  detail  formation  of  the  letters. 

^6.     The  horizontal  and  slanting  guide  lines  are  drawn 
exactly  in  the  same  man- 
ner  as  for  the  style  previ-     ^or^zonrc7^^^  ^Honzonta/ 

ously    described,    and    if 

not  followed,  the  results   will  be  similar.      See  the  uneven 

appearance  of  the  second  word  in  Fig.  2(3. 

37.     By  studying  the  formation  of  the  letters  carefully, 
it  will  be  found  that  many  of  them  are  formed  on  the  same 
/       /  principle,  as  shown  in  Fig.  27. 

a  P  a^  C^O  r^^^  Q^.^jg  ^f  ^j^g  letters  a,  b,  d, 

^     p  g,  /,  and  q  are  formed  exactly  alike 

/"'^  and   have    a    slant    of  45°  with   the 

r'  n  /71  n  ./     /      horizontal.      These  ovals  should  be 

1/1/    1/   1/  made  a  little  wider  at  the  top  than 

'     ^  at    the    bottom.       Care    should    be 

/y/  //  taken   that   the   straight    downward 

strokes   are    made    parallel    to    the 
Fig.  27  .  ^ 

slanting  guide  lines.  The  letters  c 
and  e  are  commenced  in  the  same  way,  but  the  upper  loop 
in  e  should  be  formed  in  such  a  manner  that  its  axis  will  be 
at  an  angle  of  45°  with  the  horizontal.  The  r  is  ijiade  by 
having  the  down  stroke  parallel  to  the  slanting  guide  line 


24  GEOMETRICAL   DRAWING  §  13 

and  the  up  stroke  slightly  curved  in  the  same  way  as  in  the 
letter  n  (see  Fig.  27).  The  strokes  in  the  letters  /  and  _/" 
are  the  same,  with  the  position  of  the  hook  part  reversed. 

28.  The  capital  letters  shown  in  Fig.  28  are  formed  very 
nearly  in  the  same  manner  as  those  shown  in  Art.  23,  but 
differ  slightly  by  omitting  the  short  spurs  that  give  to  the 
letters  a  more  finished  appearance. 

In  the  capital  J/,  however,  there  is  a  decided  variation. 
The  J/  is  made  with  four  strokes,  putting  in  the  parallel 
sides   first.     The    two    other    strokes    should    join    midway 

A  y  M  w 

V  PROBLEM 

Fig.  28 

between  these  sides  and  at  a  distance  from  the  top  of 
about  i  of  the  height  of  the  letter.  These  strokes,  as  will  be 
seen,  are  straight  and  not  curved. 

29.  The  munerals  should  be  ^"  high  and  of  the  style 
shown  in  Fig.  25;  fractions  should  be  -|"  high  over  all.      In 

/P34567890 

Fig.  29 

Fig.  29  the  numerals  are  illustrated  to  a  larger  scale,  and  a 
comparison  with  the  style  shown  in  Fig.  21  will  disclose 
several  variations. 

The  loops  of  the  2,  3,  5.  6,  and  9  should  be  formed  so  that 
their  axes  will  beat  an  angle  of  45°  with  the  horizontal.  It 
will  be  noted  that  the  7  differs  widely  from  the  style  shown 
in  Fig.  21.  the  down  stroke  not  curving  but  having  a  straight 
slant  of  45°.  The  axis  of  the  0  and  the  loops  of  the  8  should 
slant  at  an  angle  of  60°. 


§  13  .    GEOMETRICAL   DkAWlNG  35 

Diligent  practice  for  a  short  time  and  careful  observation 
of  the  forms  of  letters  and  numerals,  as  shown  in  Figs.  21-29, 
will  soon  enable  the  student  to  acquire  skill  and  speed  in 
this  branch  of  drawing. 

30.  The  alphabet  shown  in  Fig.  30,  called  the  block 
letter,  is  to  be  used  for  the  large  headings  or  titles  of  plates, 
as  shown  on  the  copy  plates.  This  alphabet  is  not  to  be 
used  on  the  first  five  geometrical  drawing  plates.  The  let- 
ters and  figures  are  to  be  made  -^-^"  high  and  ^'  wide, 
except  M,  which  is  y^^"  wide,  and  W,  which  is  f"  wide. 
The  thickness  of  all  the  lines  forming  the  letters  is  y"^", 
measured  horizontally.  The  distance  between  any  two 
letters  of  a   word  is  y^g-",   except  where  A  follows  P  ox  F  \ 

ABCDEFGHI  J 
KLMNDPORS 
TUVWXYZa 
ie345B7B9D 


Fig.  30 


where  F,  W,  or  F  follows  Z;  where /follows  F,  P,  T,  V,  W, 
or  F;  where  T  and  A  are  adjacent,  or  A  and  V,  IV,  or  F 
are  adjacent;  in  this  case,  the  bottom  extremity  of  A  and 
the  top  extremity  of  P,  T,  K,  IV  are  in  the  same  vertical 
line,  etc. 

Since  these  letters  are  composed  of  straight  lines,  they 
can  be  made  with  the  T  square  and  triangle.  In  lettering 
the  title  of  the  drawing  plates,  the  student  should  draw 
six  horizontal  lines  -j\"  apart  in  lead  pencil,  to  represent  the 
thickness  of  the  letters  at  the  top,  center,  and  bottom; 
then,  by  use  of  the  triangle,  he  should  draw  in  the  width  of 


26  GEOMETRICAL   DRAWING  §  13 

the  letters  and  the  spaces  between  them  in  lead  pencil. 
Having  the  letters  all  laid  out,  he  can  very  easily  ink  them 
in.  Use  the  ruling  pen  for  inking  in  the  straight  outlines 
of  the  letters,  and  the  lettering  pen  for  rounding  the  corners 
and  filling  in  between  the  outlines.  It  is  well  to  ink  in  all 
the  perpendicular  lines  first,  next  the  horizontal  lines,  and 
then  the  oblique  lines. 


PLATES 

31.  Preliminary  Directions. — The  size  of  each  plate 
over  all  will  be  14"  X  18",  having  a  border  line  ^"  from  each 
edge  all  around,  thus  making  the  size  of  the  space  on  which 
the  drawing  is  to  be  made  13"  X  17".  The  sheet  itself  must 
be  larger  than  this  when  first  placed  upon  the  board,  so  that 
the  thumbtack  holes  may  be  cut  out;  the  extra  margin  is 
also  very  convenient  for  testing  the  pen,  in  order  to  see 
whether  the  ink  is  flowing  well  and  whether  the  lines  are  of 
the  proper  thickness. 

32,  The  first  five  plates  will  consist  of  practical  geo- 
metrical problems  which  constantly  arise  in  practice  when 
making  drawings.  The  method  of  solving  every  one  of 
these  problems  should  be  carefully  memorized,  so  that  they 
can  be  instantly  applied  when  the  occasion  requires,  without 
being  obliged  to  refer  to  the  text  for  help.  Particular 
attention  should  be  paid  to  the  lettering.  Whenever  any 
dimensions  are  specified,  they  should  be  laid  off  as  accu- 
rately as  possible.  All  drawings  should  be  made  as  neat  as 
possible,  and  the  penciling  entirely  finished  before  inking  in 
any  part  of  it.  Great  care  should  be  taken  in  distributing 
the  different  views,  parts,  details,  etc.  on  the  drawing,  so 
that  when  the  drawing  is  completed,  one  view  will  not  be 
so  near  to  another  as  to  mar  the  appearance  of  the  drawing. 
The  hands  should  be  perfectly  clean,  and  should  not  touch 
the  paper  except  when  necessary.  No  lines  should  be 
erased  except  when  absolutely  necessary;  for,  whenever  a 
line  has  once  been  erased,  the  dirt  flying  around  in  the  air 


k-      A     — 1 

k  1          I 

^ 

[ 

j     PROBLE7.i:2:  To  cLrxx. 

i                        <^^ 
1 

- 

% 

: 
1 

\ 

; 

• 
1 

i                             1 

■ 

J'RCBLETyl  3:     To  dLrcLio  a. pe-rperid-icuZar  t 

0  a  sttoLighi  linej^rom  a.pctn, 

CJL3E  I. 

;                                     C 

N 

/ 

'■----._. 

.._-''' 

[ 

■                                                  ^ 

'' 

^- 

■                                                   / 
1                                                    ■ 

■     1     ! 

DECETrTBER  88.  l896 
L 

t* — 


t-pcr-pen,cii.cLL,la,r'  to  a-  36ra,tcfh.t  ttrte  J'rom,  ocgitt>&rtpotrct  irt  th,a.t  iin-e..                        . — 

I.                                                   \                                               CASEH.                                               —  - 

j                                                                              V^ 

1                                                                           ,' 

^x 

j 

\ 

!                                                /' 

I                                            / 

i 

i 

/ 
/ 

1         \            /' 

1                                             ^v                                               ' 

/ 

!                          ^               / 
1                                ^^-c' 

/ 

.th.oxx,tit.                                   \      PROBLEM' 4:   Tfir-o-Lcgh.  cl  gtve7-i.potrLt  to  oLracuj 

IE  H.                                         1     ex.  straight  lirte  pocruLleL  to  a gFii/erLsira.igrhi  line. 

N 

i 

\                                       1                                                   ,'                                          ^  ; 

i            ;             /'                 / 

'             1            '              "' 

~~               j 

1 

jojh:^  sjniTM,  class  jv^  4529.      \ 


§  13  GEOMETRICAL  DRAWING  27 

and  constantly  falling  on  the  drawing  will  stick  to  any  spot 
where  an  erasure  has  been  made,  and  it  is  then  very  difficult, 
if  not  impossible,  to  entirely  remove  it.  For  this  reason,  all 
construction  lines  that  are  to  be  removed,  or  that  are  liable 
to  be  changed,  should  be  drawn  lightly,  that  the  finish  of 
the  paper  may  not  be  destroyed  when  erasing  them.  When 
it  is  found  necessary  to  erase  an  ink  blot  or  a  line  that  has 
been  inked  in,  only  an  ijik  eraser  or  sand  rubber  should  be 
used.  After  the  erasure  has  been  made,  the  roughened 
part  of  the  surface  of  the  paper  can  be  smoothed  by 
rubbing  with  some  hard,  smooth  substance,  as  a  piece  of 
ivory  or  the   handle   of  a  knife. 


PliATE  I 


33.  Take  a  sheet  of  drawing  paper  15"  wide  and  20"  long 
(demy  size),  and  fasten  it  to  the  board  as  previously 
described.  On  this  draw  the  outlines  of  the  size  of  the 
plate,  14"  X  18",  and  draw  the  border  line  all  around  y  from 
the  edge  of  the  outline,  leaving  the  space  inside  for  the 
drawing  13"  X  17".  When  the  word  drazving  is  used  here- 
after, it  refers  only  to  the  space  inside  the  border  lines  and 
the  objects  drawn  upon  it.  To  understand  clearly  what 
follows,  refer  to  Plate  I.  Divide  the  drawing  into  two  equal 
parts  by  means  of  a  faint  horizontal  line.  This  line  is  shown 
dotted  in  Plate  I,  above  referred  to.  Divide  each  of  these 
halves  into  three  equal  parts,  as  shown  by  the  dotted  lines; 
this  divides  the  drawing  into  six  rectangular  spaces.  These 
division  lines  are  not  to  be  inked  in,  but  must  be  erased 
tvhen  the  plate  is  completed.  On  the  first  five  plates,  space 
for  the  lettering  must  be  taken  into  account.  For  each  of 
the  six  equal  spaces,  the  lettering  will  take  up  one  or  two 
lines.  The  height  of  all  capital  letters  on  these  plates  will 
be  -jV,  and  of  the  small  letters  f  of  this,  or  ■^".  The  dis- 
tance between  any  two  lines  of  lettering  will  also  be  -jV'. 
The  distance  between  the  tops  of  the  letters  on  the  first 
line  of  lettering  and  the  top  line  of  the  equal  divisions  of 


28  GEOMETRICAL   DRAWING  §  13 

the  drawing  is  to  be-J^";  and  the  space  between  the  bottoms 
of  the  letters  and  the  topmost  point  of  the  figure  repre- 
sented on  the  drawing  within  one  of  these  six  divisions 
must  also  be  not  less  than  ^".  This  makes  a  very  neat 
arrangement,  if  the  figure  is  so  placed  that  the  outermost 
points  of  the  bounding  lines  are  equally  distant  from  the 
sides  of  one  of  the  equal  rectangular  spaces.  Consequently, 
if  there  is  one  line  of  lettering,  no  point  of  the  figure 
drawn  should  come  nearer  than  ^"  -\-  ^"  -{-  ^"  =  1-^'  to 
the  top  line  of  the  space  within  which  it  is  represented ;  or, 
if  there  are  two  lines  of  lettering,  nearer  than  ^"  +  ^" 
+  tV'+  ts"+  i"=  lA"-  The  letter  heading  for  each  figure 
on  the  first  five  plates  will  be  printed  in  heavy-faced  type 
at  the  beginning  of  the  directions  explaining  each  prob- 
lem. The  student  must  judge  for  himself  by  the  length 
of  the  heading  whether  it  will  take  up  one  line  or  two, 
and  make  due  allowance  for  the  space  it  takes  up.  This 
is  a  necessary  precaution,  because  the  lettering  should 
never  be  done  until  the  rest  of  the  drawing  is  entirely 
finished  and  inked  in. 

Problem  1. — To  bisect  a  straigrht  line= 

See  Fig.  31;  also  1  of  Plate  I. 

CoxsTRUCTiox.  —  Draw  a    straight   Ime  A  B,     3^"    long. 
With  one  extremity  .rl  as  a  center,  and  a  radius  greater  than 

one-half  of  the  length  of 
/V  the  line,  describe  an  arc 
of  a  circle  on  each  side  of 
the  given  line;  with  the 
other  extremity  ^  as  a 
,s  center,  and  the  same  ra- 
dius, describe  arcs  inter- 
secting the  first  two  in  the 
points  C  and  D.  Join  C 
and  D  by  the  line  CD, 
and  the  point  P,  where  it 
intersects  A  B,  will  be  the 


Fig.  31 


required  point ;  that  is,  A  P=  PB,  and  P\s  the  middle  point 


§  13 


GEOMETRICAL  DRAWING 


29 


p 
Fig.  33 


of  A  B.  Since  C  D  \s  perpendicular  to  A  B,  this  construc- 
tion also  gives  2i  perpendicular  to  a  straight  line  at  its  middle 
point. 

Problem  2. — To  dra>v  a  perpendicular  to  a  straiglit 
line  from  a  given  point  in  that  line. 

Note.— As  there  are  two  cases  of  this  problem,  requiring  two  figures 
on  the  plate,  the  line  of  letters  will  be  run  clear  across  both  figures,  as 
shown  in  Plate  I. 

Case  I. —  When  the  point  is  at  or  near  the  center  of  the  line. 
See  Fig.  32 ;  also  2,  Case  I, 
of  Plate  I.  ^ 

Construction.  —  Draw 
A  B  3V'  long.  Let  P  be 
the  given  point.  With  P 
as  a  center,  and  any  radius, 

as  PD.,  describe  two  short       ^ 

arcs   cutting   A  B   in    the 

points  C  and  D.     With  C 

and    D    as    centers,    and    any    convenient    radius    greater 

than  P D,  describe  two  arcs  intersecting  in  E.      Draw  PE^ 

and  it  will  be  perpendicular  to  ^  i?  at  the  point  P. 

Case  II. —  When  the  point  is  near  the  end  of  the  line.     See 

Fig.  33;  also  2,  Case  II, 
of  Plate  I. 

Draw  AB  ZV'  long. 
Take  the  given  point  P 
about  f"  from  the  end 
of  the  line.  With  any 
point  (7  as  a  center,  and 
a  radius  O  P,  describe  an 
arc  cutting  A  B  in  P 
and  D.  Draw  D  O,  and 
prolong  it  until  it  in- 
tersects the  arc  in  the 
point  C.  A  line  drawn  through  C  and  P  will  be  perpen- 
dicular to  A  B  at  the  point  P. 


Fig.  as 


30 


GEOMETRICAL   DRAWING 


§13 


Problem  3. — To  di-a^^  a  perpendicular  to  a  straight 
line  from  a  point  ^vitlioiit  it. 

As  in  Problem  :2,  there  are  two  cases. 

Case  I. —  W/ien  the  point  lies  nearly  over  the  center  of  the 
line.     See  Fig.  34;  also  3,  Case  I,  of  Plate  I. 

CoxsTRUCTiox. — Draw  A  B  3|"  long.     Let  P  be  the  given 

point.  With  /^  as  a  cen- 
ter, and  any  radius  P D 
greater  than  the  distance 
from  P  to  A  B,  describe 
an  arc  cutting  A  B  in  C 
and  D.  With  C  and  D 
as  centers,  and  any  con- 
venient radius,  describe 
short  arcs  intersecting 
in  E.  A  line  drawn 
through  P  and  E  will  be 
perpendicular  to  A  Bat  F. 

Case  n. —  Wlien  the  point  lies  nearly  over  one  etid  of  the 
line.     See  Fig.  35;  also  3,  Case  II, of  Plate  I. 

Draw  .-i^3V' long,  and 


% 


Fig.  34 


let  P  be  the  given  point. 
With  any  point  C  on  the 
line  A  B  as  a  center,  and 
the  distance  C P  as  a  radi- 
us, describe  an  arc  P E  D 
cutting  J  i?  in  £.  With£ 
as  a  center,  and  the  dis- 
tance E  P  as  a  radius, 
describe    an    arc    cutting 


V 


/ 


/ 


Fig.  35 

the  arc  PE  D  in  D.     The  line  joining  the  points  P  and  D 
will  be  perpendicular  to  A  B. 

Proble^i  4. — Througfli  a  given  point,  to  di*a-sv  a  straight 
line  parallel  to  a  giAen  sti-aiglit  line. 

See  Fig.  36;  also  4  of  Plate  I. 

CoxsTRucTiox. — Let  P  be  the  given  point,  and  A  B  the 
given  straight  line  3|^"  long.     With  i"  as  a  center,  and  any 


§  13  GEOMETRICAL   DRAWING  31 

convenient  radius,  describe  an  arc  CD  intersecting  A  B 
in  D.  With  Z?  as  a  center,  and  the  same  radius,  describe 
the  arc  PE.  With  Z>  as  a  center,  and  a  radius  equal  to  the 
chord  of  the  arc  P E,  describe  an  arc  intersecting  C Dm  C. 
A    straight    line    drawn 

through  P  and  C  will  be        £^ ^l? 

parallel  to  A  B. 

34.  These  four  prob- 
lems form  Plate  I.  They 
should   be    carefully   and    a  — 

accurately  drawn  in  with 

-'  Fig.  36 

lead-pencil  lines  and  then 

inked  in.  It  will  be  noticed  that  on  Plate  I,  and  Figs.  31 
to  36,  the  given  lines  are  light,  the  required  lines  heavy, 
and  the  construction  lines,  which  in  a  practical  working 
drawing  would  be  left  out,  are  UgJit  dotted.  This  system 
must  also  be  followed  in  the  four  plates  which  are  to  follow. 
A  single  glance  enables  one  to  see  at  once  the  reason  for 
drawing  the  figure,  and  the  eye  is  directed  immediately  to 
the  required  line. 

In  the  first  five  plates,  accuracy  and  neatness  are  the  main 
things  to  be  looked  out  for.  The  student  should  be  certain 
that  the  lines  are  oi  precisely  the  length  that  is  specified  in 
the  description.  When  drawing  a  line  through  two  points, 
be  sure  that  the  line  goes  through  the  points;  if  it  does  not 
pass  exactly  through  the  points,  erase  it  and  draw  it  over 
again.  If  a  line  is  supposed  to  end  at  some  particular  point, 
make  it  end  there — do  not  let  it  extend  beyond  or  fall  short. 
Thus,  in  Fig.  36,  if  the  line  PC  does  not  pass  through  the 
points  /*and  C,  it  is  not  parallel  to  A  B.  By  paying  careful 
attention  to  these  points,  the  student  saves  himself  a  great 
deal  of  trouble  in  the  future.      Do  not  hurry  your  zvork. 

First  ink  in  all  of  the  light  lines  and  light  dotted  lines 
(which  have  the  same  thickness);  then  ink  in  the  heavy 
required  lines  after  the  pen  has  been  readjusted.  Now  do 
the  lettering  (first  read  carefully  the  paragraphs  under  the 
head  *'  Lettering  "),  and  finally  draw  the  heavy  border  lines, 


35  GEOMETRICAL  DRAWING  §  13 

which  should  be  thicker  than  any  other  line  on  the  drawing. 
The  word  '*  Plate  "  and  its  number  should  be  printed  at  the 
top  of  the  sheet,  outside  the  border  lines,  and  midwa)-  of  its 
length,  as  shown.  The  student's  name,  followed  b}^  the 
words  "Class"  and  "No.,"  and  after  this  his  Course  letter 
and  class  number  should  be  printed  in  the  lower  right-hand 
comer  below  the  border  line,  as  shown.  Thus,  John  Smith, 
Class  No.  C  4529.  The  date  on  which  the  drawing  was 
completed  should  be  placed  in  the  lower  left-hand  corner, 
below  the  border  line.  All  of  this  letterings  is  to  be  in  capi- 
tals ^"  high.  Erase  the  division  lines,  and  clean  the  draw- 
ing by  rubbing  ver\'  gently  with  the  eraser.  Care  must  be 
exercised  when  doing  this,  or  the  inked  lines  will  also  be 
erased.  It  is  best  to  use  a  so-called  *  *  Sponge  Rubber  "  for 
this  purpose,  as  it  will  not  injure  the  inked  lines.  If  any 
part  of  a  line  Jtas  been  erased  or  weakened,  it  must  be  redrawn. 
Then  write  with  the  lead  pencil  your  name  and  address  in 
full  on  the  back  of  5-our  drawing,  after  which  put  5'our 
drawing  in  the  empty  tube  which  was  sent  you,  and  send  it 
to  the  Schools. 


HEsTS    FOR   PLATE    I 

35.  Do  not  forget  to  make  a  distinction  bet'ween  tlie  width 
of  tJie  given  and  required  lines^  nor  forget  to  make  t/te  con- 
struction lines  dotted. 

JVhen  drawing  dotted  lines,  take  pains  to  have  tlie  dots  and 
spaces  uniform  in  length.  Make  the  dots  about  ^"  long  and 
the  spaces  only  about  one-third  tlie  length  of  the  dots. 

Try  to  get  tlie  work  accurate.  The  constructions  must  be 
accurate,  and  all  lines  or  figures  sliould  be  drawn  of  tlie  length 
or  size  previously  stated.  To  this  end.,  ivork  carefully  and 
keep  tlu  pencil  leads  I'cry  sliarp,  so  tliat  tlie  lines  will  be  fine. 

The  lettering  on  tlu  first  few  plates,  as  well  as  on  tlu  succeed- 
ing plates,  is  fully  as  important  as  the  drawing,  and  should  be 
done  in  tlu  neatest  possible  manner.  Drawings  sent  in  for 
correction  with  the  lettering  omitted  "will  be  returned  for 
completion. 


13 


GEOMETRICAL   DRAWING 


33 


The  reference  letters  like  A,  B,  C,  etc.,  as  shozvn  in  Fig.  31, 
are  not  to  be  put  on  the  plates. 

Do  not  neglect  to  trim  the  plates  to  the  required  size.  Do 
not  punch  large  Iioles  in  the  paper  with  the  dividers  or  com- 
passes. Remember  that  the  division  lines  are  to  be  erased — 
)iot  inked  in 

PLATE  II 

36.  Draw  the  division  lines  in  the  same  manner  as 
described  for  Plate  I.  The  following  five  problems  (5  to  9, 
inclusive)  are  to  be  drawn  in  regular  order,  as  was  done  in 
Plate  I,  with  problems  from  1  to  4.  The  letter  headings  are 
given  in  heavy-faced  type  after  the  problem  number. 

Problem  5. — To  bisect  a  given  angle.* 

Case  I. —  /  Vhen  the  sides  intersect  within  the  liiftits  of  the 
drazving.      See  Fig.  37. 

Construction.  —  Let 
A  0  B  he  the  angle  to  be 
bisected.  Draw  the  sides 
OA  and  OB  U"  long.  With 
the  vertex  O  as  a  center, 
and  any  convenient  radius, 
describe  an  arc  D£  inter- 
secting O  A  at  D  and  O  B 
at  E.  With  D  and  E  as 
centers,  and  a  radius  greater 
than  the  chord  of  half  the  arc  DE,  describe  two  arcs  inter- 
secting at  C.  The  line  drawn  through  C  and  O  will  bisect 
the  angle  ;  that  is,  A  O  C  =  C  O  B. 

Case  II. —  When  the  sides  do  not  i)itersect  zvitJiin  the  limits 
of  the  drawing.     See  Fig.  38. 

Construction. — Draw  two  lines,  A  B  and  CD.,  each  3i" 
long,  and  inclined  towards  each  other  as  shown.     With  any 

*  Since  the  letter  heading  in  this  problem  is  very  short,  it  will  be 
better  to  place  it  over  each  of  the  two  cases  separately,  instead  of  run- 
ning it  over  the  division  line,  as  was  done  with  the  long  headings  of  the 
two  cases  in  Plate  I.  Put  Case  I  and  Case  II  under  the  heading,  as  in 
the  previous  plate. 


Fig.  37 


M.   E.     I 


3i 


GEOMETRICAL  DRAWING 


13 


point  -£  on  C  D  a.s  a.  center  and  any  convenient  radius, 
describe  arc  F I G  H\  with  6^  as  a  center  and  same  radius, 
describe  arc  H LE F,  intersecting/^/ 6^//^ in  HsindF.  With 
Z  as  a  center  and  same  radius,  describe  arc  K GJ-  with  /as 


Fig.  38 

a  center  and  same  radius,  describe  arc  / E  K,  intersecting 
K  GJ  in  K  and  J.  Draw  HE  and./  K\  they  intersect  at  O, 
a  point  on  the  bisecting  line.  With  (9  as  a  center  and  the 
same  or  any  convenient  radius,  describe  an  arc  intersecting 
A  B  and  C  D  xn  M  and  A'.  With  M  and  .A'  as  centers  and 
any  radius  greater  than  one-half  J/ A',  describe  arcs  inter- 
secting at  P.  A  line  drawn  through  O  and  P  is  the  required 
bisecting  line. 

Problem  6. — To  divide  a  sriven  stitiiarlit  Hue  into  any 
required  nvmiber  of  equal  pait*. 

See  Fig.  39  {a). 


f< 


^' 


9-' 


Construction.  —  A  B  is 
the  given  line  ^Yt'  loi^g-  ^'^ 
is  required  to  divide  it  into 
eight  equal  parts.  Through 
one  extremity  A  of  the  line, 
draw  an  indefinite  straight 
line  A  C^  making  any  angle 
with  A  B.  Set  the  dividers 
to  any  convenient  distance,  and  space  off  eight  equal  divi- 
sions on  A  C,  as  A  K,  K I,  IH,  etc.     Join  C  and  B  by  the 


p    o 

Fig.  39  (n) 


-V     M 


13 


GEOMETRICAL  DRAWING 


35 


\c 


p. 


E. 


F.- 


K. 


R 


O      N     M     L 


Straight  line  C  B,  and  through  the  points  D,  E,  F,  G,  etc. 
draw  lines  DL,  /i  M,  etc.  parallel  to  CB,  by  using  the  two 
triangles;  these  parallels  intersect  A  B  in  the  points  Z.,  M,  N, 
etc. ,  which  are  equally  distant  apart.  The  spaces  L  M,  M N, 
N  O,  etc.  are  each  equal  to^  A  B.  Proceed  in  a  similar  way 
for  any  number  of  equal  parts  into  which  A  B  is  to  be 
divided. 

Another  method  is  shown  in  Fig.  39  (d).  Draw  A  B  a.s 
before,  and  erect  the  perpendicular  B  C.  Now  divide  the 
length-  of  y^  i?  by  the  number  denoting  the  number  of  equal 
parts  into  which  A  B  is  to  be  divided,  obtaining,  in  this  case, 
3tV'  -^  8  =  tVs"-  As  y^  C  is 
longer  than  A  B,  the  equal 
divisions  A  K,  K I,  etc.  are 
longer  than  A  T,  T R,  etc. 
and  may  be  made  any  con- 
venient length  greater  than 
A  B  ^  S.  In  this  case,  ^"  is 
the  most  convenient  frac- 
tion nearest  to  and  greater 
than  yVs";  hence,  consider  A  K,  K I,  etc.  to  be  each  y  in 
length,  thus  making  the  length  of  y^  C  8  X  i"  =  4".  With  A 
as  a  center  and  a  radius  equal  to  4",  describe  an  arc  cut- 
ting B  C  in  C,  and  draw  A  C.  Then  with  a  scale  lay  off 
A  K=  K I  =  etc.  =  ^",  and  project  K,  I,  H,  etc.  upon  A  B, 
in  T,  R,  P,  etc.,  the  required  points.  The  advantage  of  this 
method  over  the  other  is  that  the  T  square  and  triangle  can 
be  used  throughout,  thus  making  it  very  much  easier  to 
draw  the  parallels  D  L,  EM,  etc. 

The  student,  when  drawing  this  plate,  is  at  liberty  to  use 
either  of  the  two  methods  given  in  this  problem. 

Problej[  7. — To  clraAV  a  straijyht  line  throxijyli  any 
given  point  on  a  f?iven  straight  line  to  make  any 
required  angle  with  that  line. 

Construction. — In  Fig.  40,  A  B  \^  the  given  line  3^'  long, 
Pis  the  given  point,  and  E  O F  \s  the  given  angle.  With 
the  vertex  O  as  a  center,  and  any  convenient  radius,  describe 


Fig.  39  (3) 


36 


GEOMETRICAL   DRAWING 


Id 


Fig.  40 


an  arc  EF  cutting  O E  and  O F  m  E  and  F.     With  Pas  a 

center,  and  the  same 
radius,  describe  an 
arc  CD.  With  D  as 
a  center,  and  a  radius 
equal  to  the  chord  of 
the  arc  E  F,  describe 
an  arc  cutting  CD 
in  C.  A  line  drawn 
through  the  points  P 
and  C  will  make   an 

angle  with  A  B  equal  to  the  angle  O,  or  C PD  =  E  OF. 

Problem  8. — To  di*aTr  an  equl- 
latei*al  triangle,  one  side  being 
given. 

Construction. — In  Fig.  41,  A  B  is 
the  given  side  'IV'  long.  With  A  B 
as  a  radius,  and  A  and  B  as  centers, 
describe  two  arcs  intersecting  in  C 
Draw  C  A  and  C  B,  and  C  A  Z?  is  an  ^ 
equilateral  triangle. 

Problem  9. — The  altitude  of  an  eqiiilatei-al  triangle 
being  given,  to  di-aw  tlie  triangle. 

Construction. — In  Fig.  42,  A  B  is  the  altitude  2^"  long. 

Through  the  extremities  of  A  B  draw  the  parallel  lines  CD 

and  E  F  perpendicular  to 
A  B.  With  B  as  a  center,  and 
any  convenient  radius,  de- 
scribe the  semicircle  C  H K  D 
intersecting  CD  in  C  and  D. 
With  C  and  D  as  centers, 
and  the  same  radius,  describe 
arcs  cutting  the  semicircle 
Fi^'  -12  in  H  and  K.     Draw  B  H  and 

B  K,  and  prolong  them  to  meet  E  F  \n  E  and  F.     B  E  F  is 

the  required  equilateral  triangle. 

This  problem  finishes  Plate  II.     The  directions  for  inking 

in,  lettering,  etc.  are  the  same  as  for  Plate  I. 


Fig.  41 


c 

B 

,I> 

I 

/ 

\ 

\ 

J 

\ 

vV 

\ 

/ 

-:/ 

\. 

§13 


GEOMETRICAL   DRAWING 


37 


PLATE    III 

37.     This  plate  is  to  be  divided  up  like  Plates  I  and  II, 
and  the  six  following  prob- 
lems are  to  be  drawn  in  a 
similar  manner: 

Problem  10. — T^vo 
sides  and  the  included 
tingle  of  a  triangle  being 
given,  to  construct  the 
triangle. 

Construction. — In  Fig. 
43,  make  the  given  sides 
MNU"  long  and  PQ  1^" 

long.  Let  O  be  the  given  angle.  Draw  A  B,  and  make  it 
equal  in  length  to  PQ.  Make  the  angle  C B  A  equal  to  the 
given  angle  O,  and  make  C  B  equal  in  length  to  the  line  AI N'. 
Draw  C A,  and  C A  B  is  the  required  triangle. 


Fig.  43 


Problem  11. — To  draAv  a  i>arallelograni  ^vhen  the  sides 
and  one  of  the  angles  are  given. 

Construction. — In   Fig.  44,   make  the  given  sides  M N 
%\"  long  and  PQ  1^''   long.      Let    O    be  the  given   angle. 

Draw  A  B  equal  to  M N, 
and  draw  B  C\  making 
an  angle  with  A  B  equal 
to  the  given  angle  O. 
Make  B  C  equal  to  PQ. 
With  (T  as  a  center,  and 
a  radius  equal  to  M N., 
describe  an  arc  at  D. 
With  yi  as  a  center,  and 
a  radius  equal  to  P  Q, 
describe  an  arc  inter- 
Draw    A  D   and    CD,    and 


Fig.  44 


secting    the    other    arc    in   D 

A  BCD  is  the  required  parallelogram. 


38 


GEOMETRICAL   DRAWING 


S  13 


K- 


FIG.  45 


Problesi  1"2. — An  arc  and  its  radius  being  given,  to 
find  tlie  center. 

CoxsTRUCTiox. — In  Fig.  45,  A  C  D  B  is  the  arc,  and  J/.V, 

If"  long,  is  the  radius.  With 
MX  as  a  radius,  and  any 
point  C  in  the  given  arc  as  a 
center,  describe  an  arc  at  O. 
With  any  other  point  D  in  the 
given  arc  as  a  center,  and  the 
same  radius,  describe  an  arc 
intersecting  the  first  in  O. 
O  is  the  required  center. 
Problem  13. — To  pass  a  circumference  tlirougli  any 
tlxi'ee  points  not  in  tlie  same  straiglit  line. 

Construction. — In  Fig.  46,  A,  B,  and  C  are  the  given 
points.  With  A  and  B  as  centers,  and 
any  convenient  radius,  describe  arcs  in- 
tersecting each  other  in  A'and  /.  With 
B  and  C  as  centers,  and  any  convenient 
radius,  describe  arcs  intersecting  each 
other  in  D  and  E.  Through  /  and  K 
and  through  D  and  E,  draw  lines  inter- 
secting at  O.  With  O  as  a  center,  and 
O  A  as  a  radius,  describe  a  circle;  it  will 
pass  through  A ,  B',  and  C.  fig.  46 

Problem  14. — To  inscribe 
a  square  in  a  given  circle. 

CoNSTRUCTiox. — In  Fig.  47, 
the  circle  A  BCD  is  dl"  in 
diameter.  Draw  two  diam- 
eters, A  C  and  DB,  at  right 
angles  to  each  other.  Draw 
the  lines  A  B^  B  C,  CD,  and 
DA,  joining  the  points  of  in- 
tersection of  these  diameters 
with  the  circumference  of  the 
circle,  and  the}-  will  be  the 
sides  of  the  square. 


13 


GEOMETRICAL   DRAWING 


39 


Problem  15 — To  inscribe  a  regular  hexagon  in  a  given 
circle. 

Construction. — In  Fig.  48,  from  (9  as  a  center,  with  the 
dividers  set  to  If",  describe 
the  circle  A  B  C D  EF.  Draw 
the  diameter  D  O  A,  and  from 
the  points  D  and  A,  with  the 
dividers  set  equal  to  the  radius 
of^the  circle,  describe  arcs  in- 
tersecting the  circle  at  £,  C,  F, 
and  B.  Join  these  points  by 
straight  lines,  and  they  will 
form  the  sides  of  the  hexa- 
gon. This  problem  completes 
Plate  III.  Fig. 


PLATE  IV 

38.  The  first  four  problems  on  this  plate  are  more  diffi- 
cult than  any  on  the  preceding  plates  and  will  require  very 
careful  construction.  All  the  sides  of  each  polygon  must  be 
of  exactly  the  same  length,  so  that  they  will  space  around 
evenly  with  the  dividers.  The  figures  should  not  be  inked 
^  in  until   the   pencil    construc- 

tion is  done  accurately.  The 
preliminary  directions  for  this 
plate  are  the  same  as  for  the 
preceding  ones.      '' 

\B  Problem  16. — To  inscribe 
a  regular  pentagon  in  a 
given  circle. 

Construction. — In  Fig.  49, 
from  (?  as  a  center,  with  the 
dividers  set  to  If",  describe 
the  circle  A  BCD.  Draw  the 
two  diameters  A  C  and  D  B  at  right  angles  to  each  other. 
Bisect  one  of  the  radii,  as  0  B,  at  /.  With  /  as  a  center, 
and  /A  as  a  radius,  describe  the  arc  A/  cutting  DO  at  /. 


40 


GEOMETRICAL  DRAWING 


§13 


Wither/  as  a  center,  and  A/ as  a  radius,  describe  an  arc/// 
cutting  the  circumference  at  H.  The  chord  A  H  is,  one  side 
of  the  pentagon. 

Problem  IT. — To  inscribe  a  i*ea:iilar  octaaron  in  a  ariven 
circle. 

CoKSTRUCTiox. — In  Fig.  50,  from  6>  as  a  center,  with  the 

dividers  set  to  1|",  describe 
the  circle  ABCDEFG H. 
Draw  the  two  diameters  A  E 
and  G  C  2lX.  right  angles  to  each 
other.  Bisect  one  of  the  four 
equal  arcs,  as  A  G  at  H,  and 
'  ^  draw  the  diameter  HOD.  Bi- 
sect another  of  the  equal  arcs, 
as  A  C  at  B,  and  draw  the 
diameter  EOF.  Straight  lines 
drawn  from  A  to  B^  from  B 
to  C,  etc.  will  form  the  re- 
quired octagon. 

Problem   Is. — To  irLsci'i1>e  a  regxQar  polygon  of  any 
nuBiber  of  sides  in  a  given,  circle. 

CoxsTRUCTiON-. — In  Fig.  51,  from  (9  as  a  center,  with  the 
dividers  set  to  If",  describe  the  cir- 
cle ^  7  CD.  Draw  the  two  diam- 
eters D  7  and  A  C  at  right  angles 
to  each  other.  Divide  the  diam- 
eter D  7  into  as  many  equal  parts 
as  the  poh-gon  has  sides  (in  this 
case  seven).  Prolong  the  diam- 
eter A  C  and  make  3' A  equal  to 
three-fourths  of  the  radius  O A. 
Through  -5'  and  2^  the  second  di- 
vision from  D  on  the  diameter  D  7, 
draw  the  line  31,  cutting  the  cir- 
cumference at  /.  Draw  the 
chord  D  /,  and  it  is  one  side  of  the  required  polj-gon. 
others  may  be  spaced  off  around  the  circumference. 


Fi-: 


The 


§  13 


GEOMETRICAL   DRAWING 


41 


Problem  19. — The  side  of  a  regular   polygon   being 
given,  to  construct  the  polygon. 

Construction.— In  Fig.  o-i,  let  .-^  T  be  the  given  side.  If 
the  polygon  is  to  have  eight  sides,  the  line  A  C  should  be, 
for  this  plate,  l^^"  long.  Produce 
A  C  to  B.  From  C  as  center, 
with  a  radius  equal  to  C  A,  de- 
scribe the  semicircle  A  123  J^ 5  6 
7 B,  and  divide  it  into  as  many- 
equal  parts  as  there  are  sides  in 
the  required  polygon  (in  this 
case  eight).  From  the  point  C, 
and  through  the  second  division 
from  B,  as  6,  draw  the  straight 
line  C6.  Bisect  the  lines  A  C 
and  C6  by  perpendiculars  intersecting  in  O.  From  O 
as  a  center,  and  with  (9  T  as  a  radius,  describe  the  cir- 
cle C  A  HGFE D6.  From  C,  and  through  the  points  1,  2, 
3,  Jf,  5  in  the  semicircle,  draw  lines  C H,  C G,  C F,  etc.,  meet- 
ing the  circumference.  Joining  the  points  6  and  D,  D  and 
E,  E  and  F,  etc.,  by  straight  lines,  will  complete  the  required 
polygon. 

Problem   20. — To    find    an  arc    of  a  circle   having  a 

known   radius,  which  shall  he  equal   in   length  to   a 

given  straight  line. 

Note. — There  is  no  exact 
method,  but  the  following 
approximate  method  is  close 
enough  for  all  practical  pur- 
poses, when  the  required  arc 
does  not  exceed  \  of  the  cir- 
cumference. 

Construction.  —  In 
Fig.  53,  let  AC  he  the 
given  line  3|"  long.  At 
A,  erect  the  perpendicu- 
lar A  O,  and  make  it 
'i7  equal  in  length  to  the 
given  radius,  say  4"  long. 


Fig.  m 


4:2  GEOMETRICAL   DRAWING  §  13 

With  O  A  as  a  radius,  and  (9  as  a  center,  describe  the 
arc  A  BE.  Divide  A  C  into  four  equal  parts.  A  D  being  the 
first  of  these  parts,  counting  from  A.  With  i9  as  a  center, 
and  a  radius  DC,  describe  the  arc  C B  intersecting  ABE 
in  B.  The  length  of  the  arc  A  B  very  nearly  equals  the 
length  of  the  straight  line  A  C. 

Problem  21. — An  arc  of  a  circle  being:  given,  to  find  a 
straight  line  of  tlie  same  length. 

^-.^  This  is  also  an  approximate 

method,  but  close  enough  for 
practical  purposes,  when  the 
arc  does  not  exceed  \  of  the 
circumference. 

Construction. — In  Fig.  54, 
let  A  B  be  the  given  arc;  find 
the  center  O  of  the  arc,  and 
draw  the  radius  O  A .  For  this 
problem,  choose  the  arc  so  that 
the  radius  will  not  exceed  If".  At  A,  draw  A  C  perpen- 
dicular to  the  radius  (and,  of  course,  tangent  to  the  arc). 
Draw  the  chord  A  B,  and  prolong  it  to  D,  so  that  A  D  =  ^ 
the  chord  A  B.  With  Z>  as  a  center,  and  a  radius  D  B, 
describe  the  arc  B  C  cutting  A  C  in  C.  AC  will  be  very 
nearly  equal  to  the  arc  A  B. 


^^^ 

A 

y 

^\; 

"'^^\  / 

/ 

/ 

( 

< 

> 

\ 
\ 
\ 

FIG.  54 


PLATE  V 

39.  On  this  plate  there  are  five  problems  instead  of  six. 
It  should  be  divided  into  six  equal  parts  or  divisions,  as 
the  previous  ones.  The  two  right-hand  end  divisions  are 
used  to  draw  in  the  last  figure  of  Plate  V,  which  is  too  large 
to  put  in  one  division. 

Problem  22. — To  cli-aw  an  egg-shaped  oval. 

Construction. — In  Fig.  55,  on  the  diameter  A  B.  which 
is  2f "  long,  describe  a  circle  A  C  B  G.     Through  the  center  O, 


13 


GEOMETRICAL   DRAWING 


43 


draw  O  C  perpendicular  to  A  B^  cutting  the  circumfer- 
ence ACBG  in  C.  Draw  the  straight  lines  BCF  and 
ACE.  With  B  and  A  as  centers, 
and  the  diameter  y^  i5  as  a  radius, 
describe  arcs  terminating  in  D 
and  //,  the  points  of  intersection 
with  /)  F  and  A  E.  With  (T  as  a 
center,  and  CZ>  as  a  radius,  de- 
scribe the  arc  DH.  The  curve 
A  D  H B  G  is  the  required  oval. 

Problem  23.  —  To  draw  an 
ellii>se,  the  diameters  being 
<?iven.     The  exact  method. 


Fig.  53 


Construction. — In  Fig.  56,  let 
B  D,  the  long  diameter,  or  major 
axis,  which  is  3V'  long,  and  A  C,  the  short  diameter,  or 
minor  axis,  which  is  sy  long,  intersect  at  right  angles 
to  each  other  in  the  center  O,  so  that  D  O  =  OB  and  A  O 
=  O  C.  With  6>  as  a  center,  and  (9  C  as  a  radius,  describe 
a  circle;  with  the  same  center,  and  O  D  as  a.  radius,  describe 
another  circle.  Divide  both  circles  into  the  same  number  of 
equal  parts,  as  1-2,  2-3,  etc.  This  is  best  done  by  first 
dividing  the  larger  circle  into  the  required  number  of  parts, 

beginning  at  the  center  line 
A  C,  and  then  drawing  radial 
lines  through  the  points  of 
division  on  this  circle,  to  the 
center  O  of  the  circles,  as 
j^  shown  in  the  upper  right-hand 
quarter  of  the  figure.  The 
radial  lines  will  divide  the 
smaller  circle  into  the  same 
number  of  parts  that  the  larger 
one  has  been  divided  into. 
Through  the  points  of  division 
on  the  smaller  circle,  draw  horizontal  lines,  and,  through 
the  points  of  division  on  the  larger  circle,  draw  vertical 


Pig,  56 


44 


GEOMETRICAL   DRAWING 


§13 


lines;  the  points  of  intersection  of  these  lines  are  points  on 
the  ellipse.  Thus,  the  horizontal  line  Sc  and  the  vertical 
line  3  c  intersecting  at  c  give  the  point  c  of  the  ellipse. 
Trace  a  curve  through  the  points  thus  found  by  placing  an 
irregular  curve  on  the  drawing  in  such  a  manner  that  one 
of  its  bounding  lines  will  pass  through  three  or  more  points, 
judging  with  the  eye  whether  the  curve  so  traced  bulges 
out  too  much  or  is  too  flat.  Then  adjust  the  curve  again, 
so  that  its  bounding  line  will  pass  through  several  more 
points,  and  so  on,  until  the  curve  is  completed.  Care 
should  be  taken  to  make  all  changes  in  curvature  as  gradual 
as  possible,  and  all  curves  drawn  in  this  manner  should  be 
drawn  in  pencil  before  being  inked  in.  It  requires  con- 
siderable practice  to  be  able  to  draw  a  good  curved  line  in 
this  manner  by  means  of  an  irregular  curve,  and  the  general 
appearance  of  a  curve  thus  drawn  depends  a  great  deal  upon 
the  student's  taste  and  the  accuracy  of  his  eye. 

Problem  24. — To  draw  au  ellipse  by  circular  arcs. 

This  is  not  a  true 
ellipse,  but  is  very 
convenient  for  many 
purposes. 

Construction.  — In 
Fig.  57,  use  the  same 
dimensions  as  before. 
On  the  major  axis  A  B, 
set  ofl:  A  a=:  C  D,  the 
minor  axis,  and  divide 
a  B  into  three  equal 
parts.  With  O  as  a 
center,  and  a  radius 
equal  to  the  length  of 
two  of  these  parts,  describe  arcs  cutting  A  B  in  d  and  d'. 
Upon  i^d'  as  a  side,  construct  two  equilateral  triangles  </ (^  ^/' 
and  db'd'.  With  (^  as  a  center,  and  a  radius  equal  to  d  B, 
describe  the  arc  ^Z>/ intersecting  d  df  a.nd  d  d ' g- in  /  and  g. 
With  the  same  radius,  and  b'  as  a  center,  describe  the  arc^  Ce 


^ — ■ 

/ 

"t — ---..^^ 

*/>'^^^^ 

' 

^^■^fj 

f  \/ 

1                v 

.,  V'  ^ 

\ 

o 

A      / 

Fig.  57 


§13 


GEOMETRICAL   DRAWING 


45 


intersecting-  b'  d'  c  and  b'  d c  in  c  and  e.  With  A  and  ^  as 
centers,  and  a  radius  equal  to  the  chord  of  the  arcs  A  cox 
Be,  describe  arcs  cutting  A  B  very  near  to  d'  and  d.  From 
the  points  of  intersection  of  these  arcs  with  A  B  2i^  centers, 
and  the  same  radius,  describe  the  arcs  ^^^and  e B f. 

Problem  25. — To  clraAv  a  parabola,  tlie  axis  and  long- 
est double  ordinate  being  given. 

Explanation. — The  curve  shown  in  Fig.  58  is  called  a 
parabola.     This  curve  and   the  ellipse  are  the   bounding 


Fig.  58 

lines  of  certain  sections  of  a  cone.  The  line  O  A,  which 
bisects  the  area  included  between  the  curve  and  the  line  B  C, 
is  called  the  axis.  Any  line,  B  A  or  A  C,  drawn  perpendic- 
ular to  O  A,  and  whose  length  is  included  between  O  A  and 
the  curve,  is  called  an  ordinate.  Any  line,  as  B  C,  both  of 
whose  extremities  rest  on  the  curve,  and  is  perpendicular  to 
the  axis,  is  called  a  double  ordinate.  The  point  O  is 
called  the  A-ertex. 

Construction. — Make  the  axis  O  A  equal  to  3|",  and  the 
longest  double  ordinate  BC  equal  to  3".  B  A,  of  course, 
equals.]  C.      Draw  D Ii  through  tlie  other  extremity  of  the 


46  GEOMETRICAL  DRAWING  §13 

axis  and  perpendicular  to  it;  also  draw  B D  and  C E  par- 
allel to  O  A  and  intersecting  DE'va.  D  and  E.  Divide  DB 
and  A  B  into  the  same  number  of  equal  parts,  as  shown  (in 
this  case  six) ;  through  the  vertex  O,  draw  0 1,  O  2,  etc.  to 
the  points  of  division  on  D  B,  and  through  the  correspondii^ 
points  /,  2,  etc.,  on  A  B^  draw  hues  parallel  to  the  axis.  The 
points  of  intersection  of  these  lines,  tf,  b,  c,  etc.,  are  points 
on  the  curve,  through  which  it  may  be  traced.  In  a  similar 
manner,  dra?r  the  lower  half  OfghilCoi  the  curve. 

Probi-ew  'l-'-. — To  draw  a  iLelix.  tlie  pitcli  and  llie 
Atainetei*  l>emg  glveiL. 

ExpjLAXATiox. — The  helix  is  a  curve  formed  by  a  point 
moving  around  the  cylinder  and  at  the  same  time  advancing 
along  its  length  a  certain  distance;  this  forms  the  winding 
curved  line  shown  in  Fig.  59.  The  center  line  A  O,  drawn 
through  the  cylinder,  is  called  the  axis  of  the  helix,  and  any 
line  perpendicular  to  the  axis  and  terminated  by  the  helix  is 
of  the  same  length,  being  equal  to  the  radius  of  the  cylin- 
der. The  distance  B 12  that  the  point  advances  lengthwise 
during  one  revolution  is  called  the  pitcli. 

CoxsTRUCTiox. — As  mentioned  before,  this  figure  occupies 
two  spaces  of  the  plate.  The  diameter  of  the  cylinder  is  3|^", 
the  pitch  is  •2",  and  a  turn  and  a  half  of  the  helix  is  to  be 
shown.  The  rectangle  EB  E  D  is  a.  side  view  of  the  cylin- 
der, and  the  circle  /,  2",  S",  4-\  etc.  is  a  bottom  view.  It 
will  be  noticed  that  one-half  of  a  turn  of  the  helix  is  shown 
dotted ;  this  is  because  that  part  of  it  is  on  the  other  side  of 
the  cylinder,  and  cannot  be  seen.  Lines  that  are  hidden  are 
drawn  dotted.  Draw  the  axis  O  A  in  the  center  of  the  space. 
Draw  ED  ^"  long  and  4"  from  the  top  border  line ;  on  it 
construct  a  rectangle  whose  height  EB  =  3".  Take  the 
center  O  of  the  circle  ^"  below  the  point  H  on  the  axis  A  O, 
and  describe  a  circle  having  a  diameter  of  3^'\  equal  to  the 
diameter  of  the  cylinder.  Lay  off  the  pitch  from  B  to  12 
equal  to  2",  and  divide  it  into  a  convenient  number  of  equal 
parts  (in  this  case  12),  and  divide  the  circle  into  the  same 


13 


GEOMETRICAL  DRAWING 


47 


number  of  equal  parts,  beginning  at  one  extremity  of  the 
diameter  12'  O  6',  drawn  parallel  to  B  E.  At  the  point  1'  on 
the  circle  divisions,  erect  I'-l'  perpendicular  to  B  E\  through 
the  point  1  of  the  pitch  divisions,  draw  1~1'  parallel  to  B  E^ 
intersecting  the  perpendicular  in  1\  which  is  a  point  on  the 
helix.  Through  the  point  2\  erect  a  perpendicular  ^'-^', 
intersecting  2-2'  in  ^',  which  is  another  point  on  the  helix. 


So  proceed  until  the  point  6  is  reached;  from  here  on,  until 
the  point  12  of  the  helix  is  reached,  the  curve  will  be  dotted. 
It  will  be  noticed  that  the  points  of  division  7',  8\  9',  10', 
and  11'  on  the  circle  are  directly  opposite  the  points  f/,  4', 
3',  2',  and  T  ;  hence,  it  was  not  necessary  to  draw  the  lower 
half  of  the  circle,   since  the  point  5'  could  have  been  the 


48  GEOMETRICAL   DRAWING  §  13 

starting  point,  and  the  operation  could  have  been  conducted 
backwards  to  find  the  points  on  the  dotted  upper  half  of  the 
helix.  The  other  full-curved  line  of  the  helix  can  be  drawn 
in  exactly  the  same  manner  as  the  first  half. 

This  ends  the  subject  of  practical  geometry.  Mechan- 
ical drawing,  or  the  representation  of  objects  on  plane 
surfaces,  will  now  be  commenced. 


THE   EEPPiESEXTATIOX  OF   OBJECTS 

•40.  There  are  five  kinds  of  lines  used  in  mechanical 
drawing,  thus: 

The  liglit  full  Ihie 

The  dotted  line.       

The  broken-atid- 

dotted  line. 

The  broken  line. 

The  Jieavyf  till  line.  ,_^^^^^^^_^^_._._^_^_^_^_ 

The  light  full  line  is  used  the  most;  it  is  used  for  draw- 
ing the  outlines  of  figures,  and  all  other  parts  that  can  be 
seen  by  the  eye. 

The  dotted  line,  consisting  of  a  series  of  very  short  dashes, 
is  used  in  showing  the  position  and  shape  of  that  part  of  the 
object  represented  by  the  drawing  which  is  concealed  from 
the  eye  in  the  view  shown ;  for  example,  a  hollow  prism 
closed  on  all  sides.  The  hollow  part  cannot  be  seen;  hence 
its  size,  shape,  and  position  are  represented  by  dotted  lines. 

The  broken-and-dotted  line,  consisting  of  a  long  dash,  and 
two  dots  or  very  short  dashes  repeated  regularly,  is  used  to 
indicate  the  center  lines  of  the  figure  or  parts  of  the  figure, 
and  also  to  indicate  where  a  section  has  been  taken  when  a 
sectional  view  is  shown.  This  line  is  sometimes  used  for  con- 
struction lines  in  geometrical  figures. 


§  13  GEOMETRICAL   DRAWING  49 

The  broken  line,  consisting  of  a  series  of  long  dashes,  is  used 
in  putting  in  the  dimensions,  and  serves  to  prevent  the  dimen- 
sion lines  from  being  mistaken  for  lines  of  the  drawing. 

The  heavy  full  lines  are  made  not  less  than  twice  as  thick 
as  the  light  full  lines,  and  are  used  for  shade  lines. 

Further  explanations  in  regard  to  these  lines  will  be  given 
when  the  necessity  for  using  them  arises. 

41.  The  illustrations  in  this  and  the  following  paragraphs 
sliould  l)e  carefully  studied,  but  the  student  is  not  required 
to  send  in  drawings  from  same.  In  Fig.  60  is  shown  a  per- 
spective view  of  a  frustum  of  a  pyramid  having  a  rectangu- 
lar base  and  a  hole  passing  through  the  center  of  the  frustum. 
This  figure  represents  the 
frustum  as  it  actually 
appears  when  the  eye  of 
the  observer  is  in  a  certain 
position.  The  angles  at 
A,B,  C,  and  D  are  right 
angles,  the  hole  is  round, 
and  the  sides  A  B  and  D  C 
are  of  equal  lengths;  so 
also  are  A  D  and  B  C;  but,  if  they  were  measured  on  the 
drawing,  it  would  be  found  that  their  lengths  are  all  differ- 
ent. The  same  difficulty  would  be  met  with  in  trying  to 
measure  the  angles  and  edges  of  the  sides  A  B  F E,  B  F  G  C, 
etc.  The  real  length  of  any  line  can  be  found  only  by  a 
person  perfectly  familiar  with  perspective  drawing,  and  then 
only  with  great  difficulty.  Consequently,  this  method  of 
representing  objects  is  of  no  use  to  a  patternmaker,  car- 
penter, machinist,  or  engineer,  except  to  show  what  the 
object  looks  like.  In  order  to  represent  the  object  in  such  a 
manner  that  any  line  or  angle  can  be  measured  directly,  what 
is  termed  projection  dra>vinj;f,  or  ortJiograpJiic  projection, 
is  universally  employed.  In  the  perspective  drawing  shown 
in  Fig.  60,  three  sides  of  the  frustum  are  shown,  and  the 
other  three  are  hidden;  in  a  projection  drawing,  but  one  side 
is  usually  shown,  the  other  five  being  hidden. 

.1/.  E.     l'\—i 


50 


GEOMETRICAL  DRAWING 


13 


u 


F 

Fig. 


X> 


A  line  or  surface  is  projected  upon  a  plane,   by  drawing 
perpendicular  lines  from  points  on  the  line  or  surface  to  the 

plane,  and  joining  them. 

Thus,  if  perpendiculars  be 
drawn  from  the  extremities  of  a 
line,  as  A  B,  to  another  line  HK, 
as  shown  in  Fig.  61,  that  portion 
of  H K  included  between  the 
feet  of  these  perpendiculars  is 
called  the  projection  of  A  B 
upon  H  K.  Thus,  C  D  is  the  projection  of  A  B  upon  H K, 
the  point  C  is  the  projection  of  the  point  A  upon  H K,  and 
the  point  D  is  the  projection  of  the  point  B  upon  H K. 

The  projection  of  any  point  of  A  B^  as  E^  can  be  found  by 
drawing  a  perpendicular  from  Eto  H K,  and  the  point  where 
this  perpendicular  intersects 
HKis  its  projection.  In  this 
case,  the  point  /^ is  the  projec- 
tion of  the  point  E  upon  HK. 
It  makes  no  difference 
whether  the  line  is  straight 
or  curved — the  method  of 
finding  the  projection  is  exactlj'  the  same.  See  Fig.  62. 
In  a  similar  way,  a  surface  is  projected  upon  a  fiat  surface. 
Thus,  it  is  desired  to  project  the  irregular  surface  a  b  dc. 
Fig.  63,  upon  the  flat  surface  A  B  D  C.     Draw  the  lines  a  a', 

bb'  perpendicular  to  the  flat 
surface;  join  the  points  a' 
and  b\  where  these  perpen- 
diculars intersect  the  flat 
surface  A  B  D  C,  by  a 
straight  line  a  b',  and  a'  b' 
is  the  projection  of  the  line 
a  b  upon  A  B  D  C. 

In  the  same  way,  a  c' \s 
found  to  be  the  projection 
of  ac\  c' d\  the  projection  of  cd;  and  d'b',  the  pro- 
jection   of   db.       Hence,    the    projection   of   the    irregular 


H- 


F 
Fig.  62 


X) 


§  13 


GEOMETRICAL  DRAWING 


51 


Fig.  04 


surface  abdc  upon  the  flat  surface  AB DC  is  the  quadri- 
lateral a'  b'  d'c'. 

The  projection  of  any  point,  as  e,  is  found  as  before,  by 
drawing    a   perpendicular    ^ 
from  the  point  e  to  the  sur- 
face ;  thus,  e'  is  the  projection 
of  the  point  e  upon  the  plane 
ABDC. 

Suppose  that  the  frustum, 
Fig.  (30,  were  placed  on  a 
plane  surface  (a  surface  per- 
fectly flat,  like  a  surface 
plate),  and  the  outline  of  the  J^*" 
bottom  were  traced  by  pass- 
ing a  pencil  along  its  edges,  including  the  round  hole,  the 
result  would  look  like  Fig.  64,  in  which  the  rectcLngle  £  F G  N 
represents  the  bottom  of  the  frustum  and  the  circle  repre- 
sents the  hole. 

The  angles  and  lengths  of  the  sides  are  exactly  the  same 
as  they  are  on  the  frustum  itself;  a  similar  drawing  could  be 
made  to  represent  the  top,  but  it  is  unnecessary,  for  the 
reason  that  the  top  can  be  projected  on  Fig.  64,  and  both 
objects  accomplished  in  one  drawing.  Fig.  65  illustrates 
the  meaning  of  the  last  statement.  Here  A' B'  is  the 
projection  of  the  edge  A  B,  Fig.  60 ;  B'  C,  of  B  C,  etc. 
ff'  „*  A'  £'   is    the    projection   of 

the  edge  A  E;  B'  F,  oi  B F, 
etc.  This  drawing  shows 
the  figure  as  it  would  look 
if  the  eye  were  directly  over 
it.  A  drawing  Avhich  rep- 
resents the  object  as  if  it 
were  resting  on  a  horizontal 
plane,  and  the  observer 
looking  at  it  from  above, 
is  called  a  top  view,  or 
plan.  The  line  of  vision  is  thus  perpendicular  to  the 
faces  A  B  CD  and  E F G  H  oi  the  frustum.     The  lines  A  B, 


Fig.  65 


52 


GEOMETRICAL  DRAWING 


I  13 


/ 


-»p 


BC^  etc.,  EF,  FG,  etc.,  and  the  diameter  of  the  hole,  can 
be  measured  directly.  The  drawing  is  not  yet  complete, 
since  it  does  not  show  whether  the  ends  and  sides  are  romid- 
mz.  bellowed  out,  or  flat.  For  this  purpose,  two  more 
.  _  views  are  necessary — a  verti- 

cal projection,  OT  front  view. 
commonly  called  a  front 
elevation,  and  a  side  pro- 
jection^ or  side  vie^sv.  A 
front  view  (elevation)  is 
drawn  by  imagining  the  eye 
to  be  so  situated  that  the 
observer  looks  directly  at 
the  front  of  the  object;  in  other  words,  the  line  of  vision 
is  parallel  to  the  faces  of  the  frustum.  The  side  looked  at 
is  then  drawn  as  if  it  were  projected  on  a  vertical  plane 
at  right  angles  to  the  horizontal  plane,  the  vertical  plane 
being  also  parallel  to  the  edges  ^/^  and  H  G  oi  the  frustum 
shown  in  Fig.  60.  The  drawing  would  then  look  like  Fig.  66. 
Here  the  trapezoid  A  B  F E  represents  the  side  A  B  F E  of 
the  frustum;  the  altitude  of  the  trapezoid  being  the  same 
as  the  altitude  of  the  frustiun,  it  can  be  measured  directly. 
The  hole  cannot  be  seen  when  the  observer  looks  at  the 
frustum  in  this  position;  hence,  it  is  indicated  by  dotted 
lines.  The  projections  of  the  lines  A  B  and  D  C  (also,  of 
E  F  and  H  G,  of  A  E  and  D  H,  and  of  B  /"and  C  G)  coincide. 
To  draw  the  side  view  (sometimes  called  a  side  eleva- 
tioiL),  imagine  the  frustum  to  be 
revolved  around  on  its  axis  90°  to 
the  left,  and  then  draw  it  in  pre- 
cisely the  same  manner  as  the 
front  elevation,  by  projecting 
the  diflEerent  lines  upon  a  plane 
at  right  angles  to  the  horizontal 
plane,  and  perpendicular  to  the  .   _    . 

edges  EFzLndi  H  G^  that  is,  par- 
allel to  BC  2indFG.     The  side  elevation  would  then   be 
drawn  as  shown  in   Fig.   67.     In   this  view  the  lines  A  D 


§13 


GEOMETRICAL  DRAWING 


53 


and  BC  (also,  EII  and  FG,  DH  and   C  G,  and  A  E  and 
B  F)  coincide. 

42.  In  order  to  show  clearly  the  different  views,  and  to 
guard  against  one  view  being  mistaken  for  another,  they  are 
always  arranged  on  the  drawing  in  a  certain  fixed  and  invari- 
able manner.     Fig.  68  shows  this  method  of  arrangement. 


The  plan  is  drawn  first,  then  the  two  elevations.  It  is  usu- 
ally immaterial  which  of  these  views  is  drawn  first,  but  the 
general  arrangement  is  as  shown.  Any  departure  from  this 
method  of  arrangement  should  be  distinctly  specified  on  the 
drawing  in  writing,  unless  the  purpose  of  the  draftsman  is 
so  clearly  evident  that  no  explanation  is  needed.  The  broken 
and  dotted  lines  are  the  centei*  lines ;  they  serve  to  show 
the  connection  existing  between  the  different  views  of  the 
object,  and  to  indicate  axes  of  cylindrical  surfaces  of  any 


54  GEOMETRICAL  DRAWING  §  13 

kind.  It  will  be  noticed  that,  in  the  plan  view,  the  two 
center  lines  cross  each  other  at  right  angles,  and  that  their 
point  of  intersection  O  is  the  center  of  the  circle  which  rep- 
resents the  hole.  Whenever  a  circle  is  drawn,  two  center 
lines  should  also  be  drawn  through  its  center  at  right  angles 
to  each  other;  this  enables  any  one  looking  at  a  drawing  to 
instantly  locate  the  center  of  any  circle.  This  remark  also 
applies  to  ellipses,  semicircles  etc. 

To  draw  the  frustum  as  shown  in  the  last  figure,  either 
the  front  elevation  or  the  plan  is  drawn  first — whichever 
happens  to  be  more  convenient.  Suppose  the  front  elevation 
to  be  drawn  first.  Draw  the  vertical  center  line  ;«  ;/ ;  meas- 
ure the  altitude  of  the  frustum,  and  lay  it  off  on  this  line, 
locating  the  points  /  and  K;  through  these  points,  draw  the 
lines  AB  and  ^/^perpendicular  to  ;;/;;;  make  AI=z/B 
=■  \  A  B,  measured  on  the  frustum ;  also  E K  ^  KF=^  -i  E F, 
measured  on  the  frustum,  and  draw  A  E  and  B  F.  Lay  off 
the  radius  of  the  circular  hole  on  both  sides  of  the  center 
line  ;//  ;/,  and  draw  the  dotted  lines  parallel  to  ;//  ;/  through 
the  extremities  of  these  radii  to  represent  the  hole.  The 
front  elevation  is  now  complete.  To  draw  the  plan,  decide 
where  the  center  is  to  be  located  on  in ;/,  and  draw  the  hori- 
zontal center  line  p  q.  With  the  point  of  intersection  O  of 
the  two  center  lines  as  a  center,  and  with  a  radius  equal  to 
the  radius  of  the  hole,  describe  a  circle.  Through  the 
points  A^  B,  E,  and  /%  draw  indefinite  straight  lines  parallel 
to  tn  n.  On  both  sides  of  the  center  line/  q,  lay  off  on  these 
lines  D S  and  S A,  equal  to  \  D  A^  and  H R  and  RE,  equal 
to  i  HE,  both  DA  and  //"^  being  measured  on  the  frus- 
tum. Through  the  points  H,  E,  D,  and  A,  draw  the  lines 
H G,  E F,  DC,  and  A  B,  and  join  the  points  H  and 
D,  E  and  A,  F  and  B,  and  G  and  C  by  straight  lines, 
as  shown.     The  figure  thus  drawn  will  be  the  plan. 

To  draw  the  side  elevation,  prolong  the  lines  A  B  and 
E  F,  and  draw  the  center  line  /  v.  Lay  off,  on  each  side 
of  1 2',  -Ff/and  (7 G  equal  to  i  EG,  measured  on  the  frus- 
tum, and  B  X  a.nd  A'"  (T  equal  to  ^  B  C,  measured  on  the 
frustum.     Join  5  and  F,  and  T  and  G,  by  the  straight  lines 


§  13  GEOMETRICAL  DRAWING  55 

B  F  and  C  G,  and  draw  the  hole  dotted  as  in  the  front  ele- 
vation.    The  drawing  is  now  complete. 

The  student  should  have  by  this  time  a  good  idea  of  how 
simple  objects  may  be  represented  by  the  different  views 
of  a  drawing,  and  can  now  begin  on  the  next  plate. 


DRA^VTN^G  PliATE,  TITLE:    PROJECTION'S— I 

43.  In  making  actual  drawings  of  objects  when  the 
size  of  the  plate  is  limited,  it  is  usually  impossible  to  divide 
it  up  into  a  certain  number  of  parts,  as  in  the  case  of  the 
preceding  plates,  for  the  various  figures  differ  widely  in 
their  sizes.  These  drawings  should  be  so  made  that  no  part 
shall  come  nearer  than  |"  to  the  border  line,  and  the  figures 
should  be  so  arranged  as  to  present  a  pleasing  appearance  to 
the  eye,  and  not  be  scattered  aimlessly  all  over  the  drawing. 

Fig.  1  represents  a  rectangular  prism  2"  long,  14^" 
wide,  and  f"  thick.  The  prism  is  represented  as  if  it  were 
standing  on  one  of  its  small  ends,  with  the  broad  side 
towards  the  observer.  The  elevation  A  B  D  C  is  drawn 
first;  in  this  case,  it  will  be  a  rectangle  3"  X  14-".  The  top 
view,  or  plan,  FEB  A  is  next  drawn;  this  is  a  rect- 
angle ly  X  I",  the  side  A  B  being  the  projection  of  the 
front  of  the  prism,  and  the  side  F E  of  its  back.  Lastly, 
the  side  elevation  is  drawn ;  this  is  another  rectangle  B  E  H D, 
2"  X  I",  the  side  B D  representing  the  projection  of  the 
front  of  the  prism,  and  the  side  BE  corresponding  to  the 
right-hand  end  B  E  oi  the  plan. 

Fig.  2  is  a  \vedge  standing  on  one  of  its  triangular  ends. 
It  is  the  rectangle  shown  in  Fig.  1,  cut  diagonally  through 
the  corner  from  E  to  A  on  the  plan.  It  will  be  noticed 
that  the  two  elevations  are  exactly  the  same  as  in  Fig.  1, 
the  plan  showing  the  difference  between  the  two  figures. 

Fig.  3  is  another  wedge,  standing  on  one  of  its  rectangu- 
lar sides,  formed  by  cutting  through  the  prism,  in  Fig.  1, 
from  A  to  D.  The  plan  and  side  elevation  are  the  same  as 
in  Fig.  1.      Here,  the  front  elevation  shows  the  difference 


56  MECHANICAL  DRAWING  §  14 

When  constructing-  cycloidal  teeth  for  gear-wheels,  the 
diameters  of  the  describing  circles  are  usually  made  equal 
to  one-half  the  diameter  of  the  pitch  circle  of  a  gear-wheel 
having  12  teeth  of  the  same  pitch  as  those  of  the  gear- 
wheel about  to  be  made. 

Let  d'  be  the  diameter  of  the  describing  circle;  then, 

^':=l^Xi,  or./'=^.  (4.) 

Addendum  =  .3/;  root  =  .4/;  thickness  of  teeth  for  cast 
gears  is  .48/,  and  for  cut  gears  kp. 


DRA^^IXG    FOLATE,   TITLE  :     SPUR    GEAR-AVHEEES 

91.  This  plate  shows  the  halves  of  two  cast  gear- 
Tv^heels  having  cycloidal  teeth,  which  work  together,  a 
cross-section  of  each  gear  being  also  given.  The  drawing 
is  full  size,  the  wheels  not  being  shown  entire  for  want  of 
room;  to  have  done  so  it  would  have  been  necessary  to  make 
the  drawing  to  a  reduced  scale.  The  pitch  is  1  inch,  the 
number  of  teeth  in  the  large  gear  is  36,  and  in  the  small 
one  18.  The  pitch  diameter  of  the  large  wheel  is  found  by 
formula  1  to  be 

d  — '—-  =  11.46  inches,  nearlv. 

3.1416 

The  pitch  diameter  of  the  small  gear  = 

1  X  18 


3.1416 


=  5. 73  inches,  nearly. 


The  diameter  of  the  describing  circle  is  found  by  formula 
4  to  be 

d  =  .,  ^  ,,  ,  =  l.'.-il  inches. 
3.1416 

For  all  practical  purposes,  the  diameter  of  the  describing 
circle  may  be  taken  to  the  nearest  16th  of  an  inch.  For 
circular    pitches  under  ^  inch,  approximate  the  diameter  of 


PRDJEC 


jy 


riffj. 


^,^.2. 


~j — 


£  ^- 


r  iq.3 


^ig.  6. 


JA7ri,y:.J=.Y  /<.  ,S.9r 


For  notice  of  copyright,  see  pagti 


riDN5-l. 


Fig.//. 


Fig./ 2. 


nediately  following  the  title  page. 


yOM-JV  SJYT/TM.  CLASS  JY9  4:529. 


§  13  GEOMETRICAL   DRAWING  57 

off,  on  the  center  line  /////,  the  distance  yZ  equal  to  2",  and 
through  the  points  y  and  L  draw  the  two  horizontal  lines  S  e 
and  Rf,  Project  the  points  A',  /.  //,  and  G  upon  5  r,  as 
shown  by  the  dotted  lines;  and  through  the  points  of  inter- 
section of  these  dotted  lines  with  Se,  draw  the  vertical 
lines  S R^  a  b,  c d,  and  ef,  thus  completing  the  front  eleva- 
tion. To  draw  the  side  elevation,  extend  the  lines  .S'^' 
and  Rfy  and  draw  the  center  line  t  v.  Make  U  ]\'  equal 
to  1:^",  which  is  equal  to  the  distance  between  the  parallel 
sides,  and  draw  U X  and  W  V;  also,  M Z,  the  point  iM  cor- 
responding to  the  point  A.' of  the  plan. 

Fig.  7  represents  a  liexag-onal  pyramid ;  the  distance 
between  two  parallel  sides  of  the  base  is  l^^",  and  the  altitude 
is  2".  As  in  Fig.  G,  the  plan  must  be  drawn  first.  Then,  to 
draw  the  front  elevation,  lay  off  O  I  on  the  center  line  w ;/ 
equal  to  the  altitude,  and  through  /draw  the  base  line  A'  D' . 
Project  the  points  D,  E,  etc.  of  the  plan  upon  ^\'  1  >\  as 
shown  by  the  dotted  lines,  and  join  them  with  the  point  O 
by  the  straight  lines  A'O,  F'O,  E'O,  and  D'0\  these  lines 
are  the  vertical  projections  of  the  edges  of  the  pyramid;  the 
horizontal  projections  of  the  edges  are  F  O,  E  O,  D  O,  etc. 
The  side  elevation  can  be  easily  drawn,  and  does  not  require 
a  special  description,  the  length  of  the  base  B  E  he'mg  equal 
to  the  distance  between  the  parallel  sides,  or  1^". 

Fig.  8  shows  a  rivet  ^"  in  diameter,  having  a  button 
head  ly  in  diameter.  The  side  elevation  is  not  given, 
since  it  is  exactly  the  same  as  the  front  elevation.  Either  of 
the  two  views  may  be  drawn  first,  according  to  convenience. 
vSuppose  that  the  elevation  is  first  drawn.  Draw  the  center 
line  ;//  n,  and  the  line  A  B  for  the  base  of  the  head.  On  the 
center  line  lay  off  from  the  line  A  B,  or  the  base  of  the  head, 
a  point  (9,  at  a  distance  of  J|-",  the  height  of  the  head. 
With  the  compasses  set  to  a  radius  of  |-|",  and  from  a  point 
on  the  center  line  //i  n,  describe  an  arc  A  O  B,  taking  care 
to  pass  this  arc  through  the  point  O.  Lay  off  from,  and  on 
both  sides  of,  the  center  line  ;//  n  a  distance  of  j\",  or  \  of 
the  diameter  of  the  rivet,  and  draw /:  6"  and /^//.  Draw 
the  other  center  line/'ry  of  the  plan,  and  with  O  as  a  center, 


58  GEOMETRICAL  DRAWING  §  13 

and  a  radius  equal  to  the  radius  of  the  button  head,  describe 
a  circle.  "With  the  same  center,  and  a  radius  equal  to  y^", 
describe  the  dotted  circle,  the  horizontal  projection  of  the 
rivet.  The  irregular  line  C//"  indicates  that  only  a  part  of 
the  rivet  is  shown.  This  is  done  so  as  not  to  take  up  too 
much  space  on  the  drawing. 

Fig.  9  shows  an  ordinary  square-headed  bolt  y  in  diam- 
eter, having  a  head  If"  square  and  if"  thick.  DraAV  the 
center  lines  ;;/  u  and  /  g.  Construct  the  rectangle  A  B  D  C, 
If'X  \^\  the  elevation  of  the  head.  Locate  the  points  E 
and  F  2X  z.  distance  of  -^'  from  each  side  of  the  center  line, 
and  draw  E  G  and  F H.  With  the  compasses  set  to  a  radius 
of  If"  and  from  a  point  on  the  center  line  in  n,  describe  the 
arc  representing  the  chamfering  of  the  head.  Draw  the 
plan  of  the  head  LKBA  (a  square  whose  edge  meas- 
ures If"),  and  the  dotted  circle  f"  in  diameter,  the  pro- 
jection of  the  body  of  the  bolt,  which  cannot  be  seen  in 
this  view. 

Fig.  10  shows  a  distance  piece  used  to  separate  two 
other  parts,  and  to  keep  them  a  certain  distance  apart. 
The  arrangement  of  the  views  of  this  figure  is  somewhat 
different  from  the  preceding  ones,  in  order  to  make  room 
for  it  on  the  drawing.  Draw  the  center  line  ;/  in,  and  con- 
struct the  figure  according  to  the  dimensions  marked  on 
the  plate.  Use  a  radius  of  ^"  for  the  fillets  at  A,  B,  C, 
and  D,  and  an  equal  radius  to  round  the  corners  at  E, 
E,  G,  and  //. 

Fig.  11  shows  a  square  cast-iron  Tvaslier.  Instead  of 
making  an  elevation  and  plan  as  usual,  a  section  is  taken 
through  /)  q;  that  is,  the  washer  is  imagined  to  be  cut  on  the 
line/^,  with  all  that  part  of  the  figure  to  the  left  oi  p  q 
removed,  and  an  elevation  drawn  of  the  remaining  part.  In 
order  to  distinguish  a  sectional  drawing  without  any  possi- 
bility of  mistake,  the  so-called  section  lines  are  employed. 
These  are  usually  drawn  by  laying  a  45°  triangle  against  the 
edge  of  the  T  square,  and  drawing  a  series  of  parallel  lines 
as  nearly  equally  distant  apart  as  can  be  judged  by  the  eye. 
For  cast  iron,  these  lines  are  full,  thin  lines,  all  of  the  same 


§  13  GEOMETRICAL  DRAWTNCr  59 

thickness,  and  must  not  be  drawn  too  near  together.  The 
method  of  sectioning  for  other  materials  will  be  given  later 
on.  It  is  not  usual  to  draw  the  section  lines  in  pencil,  but 
to  wait  until  the  outlines  of  the  drawing  have  been  inked  in, 
and  then  section  directly  with  the  drawing  pen.  The  short- 
est distance  apart  of  the  section  lines  should  rarely  be  less 
than  -i^",  unless  the  drawing  is  of  such  small  dimensions  as 
to  cause  a  sectioning  of  this  width  to  look  coarse.  This  is 
the  case  with  Figs.  11  and  12  of  this  plate.  In  these  two 
figures  make  the  section  lines  a  full  -^^"  apart.  Only  that 
part  of  the  figure  is  sectioned  which  is  touched  by  the  cut- 
ting plane,  the  rest  of  the  figure  being  drawn  as  if  it  were 
projected  upon  the  cutting  plane.  The  corners  of  this 
figure  should  be  rounded  with  a  radius  of  yV">  the  other 
dimensions  can  be  obtained  from  the  plate. 

Fig.  12  is  a  east-iron  cylindrical  ring.  It  is  shown 
in  plan  and  section.  The  dimensions  given  suffice  for 
the  drawing  of  the  figure  without  further  explanations. 
The  inner  circle  of  plan  is  the  projection  of  the  inner- 
most points  of  the  ring  which  form  a  circle  whose  diameter 

44.  When  inking  in  a  drawing,  it  is  generally  best  to 
draw  the  circles  and  other  curved  lines  first,  and  the  straight 
lines  afterwards.  This  enables  the  draftsman  to  easily  blend 
into  one  line  the  straight  lines  meeting  the  curves,  so  that 
their  points  of  meeting  cannot  be  detected;  it  enables  the 
tangent  lines  to  be  drawn  with  better  success,  and  also 
shortens  the  time  of  inking  in  a  drawing.  It  will  be  noticed 
that  some  of  the  straight  lines  are  heavy  and  some  light, 
and  that  parts  of  the  full-line  circles  are  heavy  and  the  rest 
of  the  circle  light.  These  are  the  shade  lines;  they  are 
described  later  on.  The  student  may  make  all  of  the  full 
lines  except  the  border  lines  of  this  plate,  and  the  three  fol- 
lowing plates,  of  the  same  thickness,  if  he  so  desires.  The 
dotted  lines  used  to  indicate  those  parts  of  the  figures  that 
are  hidden  must  be  of  the  same  thickness  as  full  lines,  while 
the  construction  lines  and  center  lines  should  be  very  thin 


60 


GEOMETRICAL   DRAWING 


§13 


r=^^-_^^-.-.&lvi^ 


J 


Fig.  69 


45.  Dlmensioiis. — The  dimension  lines  and  figures  on 
this  and  succeeding  plates  are  to  be  inked  in  by  the  student. 

Make  the  dimension 
figures  y\"  high,  and  of 
the  same  style  as  those 
shown  in  Art.  20. 
Fractions  should  be  A' 
high  over  all.  If  there 
is  not  room  for  figures 
of  this  size,  great  care 
should  be  taken  to  make 
them  clear. 

Until  after  the  student  has  obtained  sufficient  practice  in 
lettering,  he  should  draw  guide  lines  in  pencil  for  the  dimen- 
sion figures,  as  in  Fig.  69,  unless  he  can  make  them  look 
well  without.  All  the  figures  should  have  the  same  slant  of 
60°,  and,  when  printing  fractional  dimensions,  the  luJiole 
fraction  should  have  the  same  slant  as  the  figures;  that  is, 
the  denominator  should  be  under  the  numerator  in  a  i'/^;///;/^ 
direction,  and  not  straight  below  it.  Make  the  dividing 
line  between  the  numerator  and  denominator  horizontal, 
not  slanting. 

Dimension  and  extension  lines  must  be  light,  broken  lines 
of  the  same  thickness  as  the  center  and  construction  lines. 
Care   should    be   exercised    to 
make  the  arrowheads  as  neatly 
as  possible  and   of  a  uniform 
size.      They  are    made  with  a 
Gillott's  No.  303  pen,  and  their 
points   must  touch  the  exten- 
sion lines,  as  illustrated  in  Fig 
heads  too  flaring. 

When  putting  in  the  dimensions,  care  should  be  taken  to 
give  <?// that  would  be  needed  to  make  the  piece  which  the 
drawing  represents,  but  do  not  repeat  the  same  dimension 
on  different  views.  Thus,  in  Fig.  1  of  this  plate,  the  length 
is  given  in  the  front  elevation  as  "2",  and  it  is  obviously 
unnecessary  to  give  the  same  dimension  in  the  side  elevation. 


Bight 


Tic-.  :■'.' 
TO.     Do  not  make  arrow 


§  13  GEOMETRICAL   DRAWIXCx  61 

Again,  the  dimension  lines  should  be  put  where  they  would 
be  most  likely  to  be  looked  for.  In  Fig.  10  of  this  plate, 
the  diameter  of  the  central  part  of  the  distance  piece  is 
marked  1^"  in  the  elevation  •  it  could  have  been  marked 
on  the  side  elevation,  as  the  diameter  of  the  dotted  circle, 
but  a  person  wishing  to  find  the  size  of  this  part  of  the  piece 
would  naturally  look  for  it  in  the  front  elevation.  This  is 
also  true  of  the  diameter  of  the  flange.  The  diameter  of 
the  hole  could  be  on  the  plan  or  elevation,  but  it  is  put 
on  the  plan  because  it  is  denoted  there  by  a  full  line,  while  in 
the  elevation  the  hole  is  dotted.  Never  cross  one  dimension 
line  by  another,  if  it  can  well  be  avoided.  Thus,  in  Figs.  2 
and  4  of  this  plate,  the  bounding  lines  of  the  triangular 
views  are  extended  by  fine  broken  lines,  in  order  that  the 
dimension  lines  (J")  may  not  cross  the  lines  marking  the 
length  and  width  of  the  wedge. 

The  student  should  ink  in  all  the  figures  used  for  dimen- 
sions shown  on  this  and  succeeding  plates,  on  his  drawing, 
but  should  omit  the  letters  used  to  describe  the  different 
objects.  The  titles  should  be  made  in  block  letters  as 
shown  on  sample  copies.  The  date,  name,  course  letter, 
and  class  number  are  to  be  put  on  as  in  the  preceding- 
plates. 


DRAWIXG  PLATE,  TITIiE :    PROJECTIONS— II 

4G.  The  figures  on  the  last  plate  were  drawn  under  the 
supposition  that  the  center  lines,  and  at  least  one  flat  side, 
were  parallel  to  the  plane  of  the  paper — the  center  lines 
were  also  either  vertical  or  horizontal.  This  is  always  pos- 
sible in  detail  drawings,  where  each  piece  is  drawn  sep- 
arately by  itself,  but  in  the  case  of  machines,  where  the 
parts  are  placed  at  different  angles,  they  cannot  always 
be  drawn  in  this  manner.  The  figures  on  this  plate  are 
so  drawn  that  they  show  objects  similar  to  those  in  the 
last  plate,  but  at  different  angles.  The  student  should 
exercise  particular   care   to  understand  this  plate  and  the 


62  GEOMETRICAL  DRAWING  §  13 

two  succeeding  ones;  if  he  thoroughly  masters  them, 
he  should  experience  no  great  difficulty  in  the  plates  that 
follow. 

Fig.  1  shows  a  rectangnlar  prism  2f '  long,  2"  wide, 
and  1"  thick,  standing  in  a  perpendicular  position  on  one  of 
its  small  ends  in  such  a  manner  that  the  broad  sides  make 
an  angle  of  30°  with  a  horizontal  line.  Draw  the  plan  first. 
To  do  this,  construct  the  rectangle  A  BCD  2"  X  1'',  with 
the  parallel  edges  A  B  and  D  C  making  an  angle  of  30°  with 
the  horizontal  ;  this  may  be  done  by  holding  the  head  of  the 
T  square  against  the  left-hand  end  of  the  board,  and  using 
the  60°  triangle.  To  construct  the  front  elevation,  draw  a 
horizontal  line  A'C  and  project  A  upon  this  line,  thus 
obtaining  the  point  A' .  Draw  A'E  perpendicular  to  A'O^ 
and  make  it  equal  in  length  to  2|^",  the  length  of  the  prism. 
Through  E  draw  E  G.  Project  the  points  B  and  C  upon 
A'C,  and  draw  BE  and  C'G.  The  back  edge  UH  of  the 
prism  is  not  seen,  and,  hence,  its  position  is  indicated  by  the 
dotted  line  UH. 

The  side  elevation  can  be  drawn  in  a  similar  manner 
by  projecting  the  points  A  BCD  upon  a  vertical  line, 
as  I L.  Produce  A'C  and  EG,  and  make  B' D  equal  to 
I L.  Now  use  the  spacing  dividers,  and  set  oflE  B'C 
equal  to  IC,  and  B'A'  equal  to  IK.  Through  B\  C\ 
A',  and  D,  draw  the  vertical  lines  B'E,  C'G,  A'E,  and 
DH,  drawing  A'E  dotted,  because,  when  looking  at  the 
prism  in  the  direction  of  the  arrow,  the  edge  A'E  is 
not  seen. 

Fig.  3  is  the  same  prism  shown  in  Fig.  1,  but  in  a  diflFerent 
position.  The  two  broad  sides  are  parallel  to  the  plane  of 
the  paper,  and  the  prism  is  tipped  in  such  a  manner  that  the 
base  makes  an  angle  of  160°  with  the  horizontal.  The  ele- 
vation must  be  drawn  first.  To  do  this,  draw  a  horizontal 
fine;  then,  by  using  the  protractor,  draw  the  line  EE, 
making  an  angle  of  160°  with  the  horizontal,  reckoning  from 
right  around  to  the  left,  opposite  to  the  motion  of  the  hands 
of  a  clock.  Make  E  F  equal  in  length  to  2",  and  on  it  con- 
struct the  rectangle  EF  B  A,  2f "  X  2" ;  it  will  be  the  vertical 


/>^. 


E        J-C 


-mmc 


^H 


PRDJEC 


i^^' 


T  Z^.^. 


J-t^r- 


^^-^  .  ■ — nL.-:,.    .— ,  -  :  jy 


:  f  coryrlgr.t.  see  pagt 


riDNS-ll. 


c 


\B' 


^' 


9  2- 


s-b: c' 


1 

1 

^ 

' 

"^r^l 

J^z.y.. 


mediately  following  the  title  page. 


JO^HT-/  S7yZITJ-£.  CZAS^JV?  45£9. 


§  13  GEOMETRICAL  DRAWING  63 

projection  or  front  elevation  of  the  prism.  The  method  of 
drawing  the  plan  and  side  elevation  is  apparent  without 
further  explanation. 

Fig-.  3  is  the  same  prism  shown  in  Figs.  1  and  2,  but  with 
the  narrow  sides  parallel  to  the  plane  of  the  paper,  and  tipped 
until  the  base  makes  an  angle  of  17^°  with  the  horizontal. 
The  sizes  are  the  same  as  in  the  two  preceding  figures,  and 
it  should  be  drawn  without  further  explanation,  the  front 
elevation  being  drawn  first. 

Fig.  4  shows  a  liexagonal  prism  having  two  of  its  par- 
allel sides  parallel  to  the  plane  of  the  paper,  and  its  axis 
vertical;  instead  of  a  side  elevation  at  right  angles  to  the 
horizontal,  a  side  elevation  is  desired,  as  if  the  vertical  prism 
were  looked  at  in  the  direction  of  the  arrow,  or  at  an  angle 
of  30"  with  the  horizontal.  Draw  the  plan  first  and  then  the 
front  elevation  from  the  dimensions  given.  To  draw  the 
other  view,  first  draw  the  center  line  vi  n,  and  then,  by  use 
of  the  T  square  and  30''  triangle,  draw  the  lines  A  B,  CD,  E  F, 
and  G  //,  from  the  points  A,  C,  E,  and  G,  as  shown.  Also 
draw  in  a  similar  manner  the  other  four  dotted  lines  at  the 
base  of  the  prism;  then  draw  the  line  IB  at  a  right  angle  to 
the  lines  A  B,  CD,  etc.  At  the  point  /,  draw  the  line  // 
parallel  to  the  center  line  ;;/  n,  and,  Avith  /as  a  center,  and  the 
points  j5,  D,  F,  H,  K,  L,  etc.  as  radii  describe  arcs,  as  shown, 
cutting  the  vertical  line  //at  the  points/,  M,  N,  O,  P,  Q, 
etc.  Through  the  points/,  M,  N,  O,  P,  Q,  etc.  draw  hod- 
zontal  lines  as  shown.  On  each  side  of  the  vertical  center 
line  mn,  lay  off  a  distance  of  f",  or  one-half  the  distance 
between  the  parallel  sides  of  the  prism,  which  is  1|",  as 
shown  in  the  plan,  and  draw  the  lines  7?  5  and  T  U.  This 
view  is  then  completed  by  drawing  the  lines  V R,  V  T,  IV X, 
and  YX,  as  shown.  The  lines  at  the  base  are  drawn  in  a 
similar  manner. 

Fig.  5  represents  a  hexagfonal  pyramid  whose  axis  is 
parallel  to  the  plane  of  the  paper,  the  base  making  an  angle 
of  30°  with  the  horizontal.  It  is  desired  to  find  the  vertical 
projection  of  the  side  elevation.  Having  drawn  the  plan 
ABCDEFam]   the  side   elevation   6''y^'/i"6'7/,  as  shown 


64  GEOMETRICAL   DRAWING  §  13 

from  the  dimensions  marked  on  the  drawing,  choose  the 
position  of  the  vertical  center  line  t'v\  project  (?'  and  (?'" 
upon  it  in  the  points  O"  and  0^^\  and,  through  O^  and  6^'", 
draw  a  fourth  center  line  r  s.  On  this,  lay  off  O^  G  and 
O"  IT  equal  to  O  G  and  O  H,  and  construct  the  projec- 
tion A"  B"  C"  D"  E"  F' ,  as  indicated  b\^  the  broken  and 
dotted  lines.  Join  O"  E" ,  O"  F',  etc.  b)-  straight  lines,  and 
it  will  be  the  required  projection.  The  figure  thus  drawn 
represents  the  pyramid  as  it  would  appear  placed  so  that 
its  base  made  an  angle  of  30°  with  the  horizon,  the  line  of 
vision  being  horizontal  to  the  observer  looking  at  it  from 
the  left  side. 

Fig.  6  shows  a  cyltader  whose  axis  is  parallel  to  the  plane 
of  the  paper  and  makes  an  angle  of  77°  with  the  horizontal. 
The  vertical  side  projection  is  required.  Draw  the  plan  and 
front  projection  as  shown  from  the  dimensions  given.  Draw 
the  center  line  t  v  vertical,  and  project  the  center  CX  upon 
it  in  (9";  also,  A  in  A\  and  H'  in  IF .  To  find  the  remain- 
ing points  on  the  projected  circle,  divide  the  diameter  A  H 
of  the  plan  into  a  convenient  number  of  equal  parts,  in  this 
case  7,  as  A  1,  1-2,  2S,  etc.  Through  the  points  thus  laid 
off,  draw  the  lines  1-1'\  2-2'\  3~S'\  etc.,  parallel  to  the  cen- 
ter line  inn.  Through  the  points  A\  T\  ^",  ^",  etc.,  draw 
the  horizontal  lines  as  shown  b}^  the  dotted  lines.  From  and 
on  each  side  of  the  vertical  center  line  /  v^  lay  off  distances 
on  each  side  of  the  horizontal  lines  just  drawn  equal  to  the 
length  of  that  part  of  the  lines  1-1'\  2-2'\  SS',  etc.  included 
between  the  center  line/^  and  the  semicircle  ^-^  CH;  thus, 
on  the  horizontal  line  drawn  through  the  point  0\  the  dis- 
tances O'  C"  and  0"D"  are  each  equal  to  (9  C  in  the  plan. 
The  distances  P^'-l'"  and  P^-P'  are  each  equal  to  the  dis- 
tance from  1  to  the  point  of  intersection  of  the  semicircle  on 
the  line  1—1".  The  remaining  distances  are  laid  off  in  a 
similar  manner.  A  curve  traced  through  the  points  thus 
found  will  be  the  required  projection  of  the  upper  base  of 
the  C5'linder.  The  projection  of  the  lower  base  is  found  in 
exactly  the  same  way.  Drawing  C"E'  and  D"F  completes 
the  required  projection. 


§  13  GEOMETRICAL  DRAWING  65 

DRAAYIXG  PLATE,  TITLE:    COIS^IC  SECTIO]S^S 

47.  This  plate  shows  the  different  forms  of  the  curves 
formed  by  the  intersection  of  a  cone  or  cylinder  by  a  plane. 
If  the  plane  of  intersection  is  perpendicular  to  the  axis  of 
the  cone  or  cylinder,  the  curve  of  the  intersection  will  be  a 
circle;  but  if  it  is  inclined  to  the  axis,  it  will  be  an  ellipse 
in  the  case  of  a  cylinder,  and  an  ellipse,  hyperbola,  or  parab- 
ola in  the  case  of  a  cone,  according  to  the  angle  of  inclination. 

Fig.  1  is  a  cone  cut  by  a  plane  which  does  not  intersect 
the  base  of  the  cone.  VVhoi  the  cutting  plane  does  not  inter' 
sect  the  base,  or  the  base  of  the  cone  extended,  the  curve  of 
intersection  is  an  ellipse. 

Draw  the  plan  and  front  elevation  of  a  right  cone  whose 
altitude  is  3f  inches  and  whose  base  is  3  inches  in  diameter. 
Cut  this  cone  by  a  plane  ab,  making  an  angle  of  52°  with 
the  base.      See  figure. 

Divide  the  circle  which  represents  the  base  of  the  cone  in 
the  plan  into  any  number  of  parts,  in  this  case  24,  and, 
through  the  points  of  division  A,  E,  H,  etc.,  draw  the 
radii  O  A,  O  E,  OH,  etc.  to  the  center  O.  Draw  also  from 
these  points  straight  lines  A  A',  E E',  H H' ,  B B\  etc.,  par- 
allel to  the  axis  of  the  cone  O'n,  and  cutting  the  base  A'B' 
in  the  points  E' ,  H',  etc.  From  these  points,  draw  lines  to 
the  apex  O'  of  the  cone,  and  cutting  the  base  A'  B'  in 
points  E',  //',  etc.  From  these  points,  draw  lines  to  the 
apex  O'  of  the  cone,  as  E'O' ,  H'O' ,  etc.,  cutting  the  plane  a  b 
in  the  points  D' ,  F ,  etc.  From  these  points  D' ,  E' ,  etc., 
draw  straight  lines  E'EE",  D' D D" ,  etc.,  parallel  to  the 
axis  O'n  of  the  cone,  and  intersecting  the  radii  O  A,  O E, 
OH,  OB,  etc.,  in  the  points  C,  D,  F,  K,  F" ,  D",  etc.,  and 
through  these  points  of  intersection  draw  the  ellipse  by  aid 
of  an  irregular  curve. 

Fig.  2  is  a  cone  of  the  same  size  as  in  the  preceding  prob- 
lem ;  but  the  cutting  plane  a  b  is,  in  this  case,  parallel  to  one 
of  the  elements*  of  the  cone,  and  intersects  the  base.      The 


*Any  straight  line  drawn  on  the  surface  of  a  cone  and  passing  through 
the  apex  (as  OH',  Fig.  1,  or  O'A',  Fig.  2,  etc.)  is  called  an  element. 


66  GEOMETRICAL   DRAWIXG  §  13 

curve  formed  by  the  intersection  of  a  cone  by  a  plane  parallel 
to  one  of  its  elements  is  called  a  parabola.  The  plan  and 
front  elevations  of  the  cone  and  curve  of  intersection  are 
found  in  a  manner  similar  to  the  method  used  in  the  last 
problem.  To  find  the  side  elevation,  proceed  as  follows: 
Draw  the  side  elevation  0"A"B"  of  the  cone  with  the  center 
line  /  ^  as  its  axis.  Draw  the  projection  lines  F*  F" F^"^^ 
D'D"'D^,  etc.,  and  make  KT"  and  K'F^"-'  equal  to  KF 
and  KF"x  make  I'D'"  and  I'D^'''  equal  to  /Z)  and  I D'\ 
etc.,  and  trace  a  curve  through  the  points  thus  found.  The 
result  will  be  the  side  elevation  of  the  cone  when  cut  by  a 
plane  parallel  to  one  of  its  elements  and  having  the  upper 
part  removed.  The  side  elevation  of  Fig.  1  may  be  drawn 
in  a  similar  manner. 

Fig.  3  is  a  cone  having  the  same  dimensions  as  the  two 
preceding  problems,  but  cut  by  a  plane  a  b  parallel  to  the  axis 
of  the  cone  and  perpendicular  to  the  vertical  plane  of  pro- 
jection. When  the  cutting  plane  intersects  the  base  of  a 
cone  and  is  not  parallel  to  any  element  (that  is,  if  the  acute 
angle  included  between  the  cutting  plane  and  the  base  is 
greater  than  the  angle  O' A'  B'  included  between  any  one 
element  and  the  base),  the  curve  of  intersection  is  called  a 
liyperbola. 

The  plan  and  front  elevation  are  constructed  as  before,  the 
horizontal  projection  of  the  curve  for  this  particular  case, 
where  the  cutting  plane  is  parallel  to  the  axis  of  the  cone,  is 
also  a  straight  line.  The  side  elevation  is  found  as  in  the 
last  problem,  by  drawing  the  lines  of  projection  F'F"'F^''\ 
j)-j)"Div^  etc.,  and  making  /'Z>'"  and  I'D'"'  equal  to  ID 
and  ID",  K'F"  and  A"/"'^  equal  to  //"and  IF",  etc.  The 
curve  drawn  through  the  points  thus  found  will  be  the 
required  hyperbola. 

Fig.  4  shows  the  intersection  of  a  cylinder,  3f "  long 
and  2"  in  diameter,  by  a  plane  a  b,  making  an  angle  of  57* 
with  the  base.  The  plan  and  elevation  may  be  drawn  as 
shown,  the  horizontal  projection  of  the  curve  being  a  circle, 
havnng  the  same  diameter  as  the  base.  To  construct  the 
side  elevation  of  the  curve,  divide  the  circle  representing  the 


^^'LT/OAHY  /4,  /39P'. 


For  notice  of  copyright,  see  pag 


>v  c« 


n 


n 

in 
n 
n 


imediately  following  the  title  page. 


^oj^jT syyr/TJ-c  cz^ss  yy9-f339. 


§  13  GEOMETRICAL  DRAWING  67 

base  of  the  cylinder  in  the  plan  into  any  number  of  parts,  in 
this  case  24,  and  through  the  points  of  division  A,  B,  C,  etc. 
draw  the  radii  OA,  OB,  O  C,  etc.  to  the  center  O.  Draw 
also  from  these  points  straight  lines  A  A,  L  B  B' ,  K  C  C, 
I D  D\  etc.  parallel  to  the  axis  ;// ;/  of  the  cylinder,  and  cut- 
ting the  base  in  the  elevation.  From  the  points  A,  B\  C, 
D\  etc.  draw  lines  D' E' ,  C'F',B'G\  etc.  at  right  angles  to 
the  axis  nui.  Make  I'E  and  /'^'each  equal  to  ID;  K'F 
and  K'F'  each  equal  to  KC\  L'G  and  L'G'  each  equal 
to  LB,  etc.  The  curve  drawn  through  these  points  Avill 
be  the  side  projection,  or  side  elevation,  of  the  curve  of 
intersection. 

DRAWING  PLATE,   TITTLE:    INTERSECTIONS  AND 
DEVELOPMENTS 

48.  On  this  plate  some  dimensions  are  given  in  decimal 
fractions  instead  of  common  fractions.  Such  decimal  dimen- 
sions should  be  laid  off  with  a  decimal  scale,  if  the  student 
has  one.  A  decimal  scale  is  a  scale  with  inches  divided  into 
tenths,  hundredths,  etc.  If  the  student  has  no  decimal  scale 
(and  such  a  scale  is  not  essential),  he  should  take  the  nearest 
value  of  the  decimal  fraction  in  thirty-seconds  of  an  inch. 

To  change  a  decimal  fraction  to  a  common  fraction,  having 
a  desired  denominator,  multiply  the  decimal  by  the  desired 
denominator  of  the  common  fraction,  and  express  the  result 
as  a  whole  number,  which  whole  number  will  be  the  numera- 
tor of  the  fraction. 

Thus,  to  express  .705"  in  fourths,  we  have  .705  X  4  =  3.()G 
fourths  ==  say,  f".  To  express  .765"  in  sixteenths,  we  have 
.7(55  X  16  =  12.24  sixteenths  =  say,  |f".  To  express  .765" 
in  thirty-seconds,  we  have  .765  X  32  =  24.48  thirty-seconds 

The  length  of  the  circumference  of  a  circle  =  the  diameter 
X  3.1416;  hence, 
The  length  of  circumference  of  a  circle  whose  diameter  is 

If"  =  3.1416  X  If"  =  4.32"  =  i^". 
The  length  of  circumference  of  a  circle  whose  diameter  is 
1{"  =  3.1416  X  U"  =  4.71"  =  4f|\ 


68  GEOMETRICAL  DRAWING  §  13 

The  length  of  circumference  of  a  circle  whose  diameter  is 

li"  =  3.1416  X  li"  =  3.93"  =  3||". 
The  length  of  circumference  of  a  circle  whose  diameter  is 
1^"  =  3.1416  X  IfV"  =  i-S"- 
4.9"  ^  2  =  2.45"  =  2r'6"  (see  Fig.  10). 

49.  This  plate  deals  with  the  intersection  of  surfaces  and 
their  development.  Fig.  1  shows  the  intersection  of  two 
unequal  cylindrical  surfaces  whose  axes  p  q  and  in  n 
intersect  at  right  angles.  Their  dimensions  are  given  in  the 
figure.  For  the  sake  of  convenience,  a  bottom  view  is  given, 
instead  of  a  top  view,  as  usual.  First  draw  the  front  elevation, 
omitting,  of  course,  the  curve  of  intersection  E  Q  G  D  C B  A, 
which  must  be  found.  Then  draw  the  side  elevation  and 
the  bottom  view,  as  shown.  Divide  the  circle  which  repre- 
sents the  side  projection  of  the  cylindrical  surface  F E  A  1 
into  any  convenient  number  of  parts,  in  this  case  12,  and 
draw  the  projection  lines  7E,6Q,5G,Ji.D,3C,  2 B,  and  1 A 
parallel  to  the  axis/^.  Also  draw  the  projection  lines  4-4', 
3-3',  2-2\  1-1' ,  etc.  parallel  to  the  axis  /  v.  Choose  a  con- 
venient point  O,  and  through  it  draw  two  lines  (9 /and  O  K 
parallel  to  the  axes  pq  and  inn  of  the  cylinders.  Continue 
the  lines  4-4',  3-3\  etc.  downwards,  until  they  cut  O  I  in  8, 
7,  6,  5,  etc.  Now  make  O 8'  =  O 8,  Ol'  —  O  1,  etc.";  this 
may  be  most  conveniently  done  by  taking  6^  as  a  center,  and 
describing  arcs  of  circles  with  radii  equal  to  O  8,  0  7,  O  6, 
etc.,  cutting  O K  in  8',  7',  6',  etc.  Through  8',  7',  6',  etc., 
draw  the  lines  8'D',  7'C',  6'B',  etc.  parallel  to  the  center 
line  r  s.  Through  the  points/^',  C',B',  and  A',  draw  the 
lines  D'D,  C'G,  and  B'Q,  parallel  to  the  center  line  ni  n,  and 
intersecting  the  lines  J/.  D,  5  G,  6  Q,  3  C,  and  ^  ^  in  the 
points  D,  G,  Q,  etc.  The  curve  traced  through  these  points 
will  be  the  front  elevation  of  the  curve  of  intersection  of  the 
two  cylindrical  surfaces. 

Fig.  2  shows  the  intersection  of  two  equal  cylindrical 
surfaces  at  right  angles  to  each  other,  as  in  the  case  of  a 
pipe  elbow.  When  two  cylinders  having  equal  diameters 
intersect,  and  their  axes  also  intersect,  the  front  elevation  of 


INTERSECTIONS  Al 


np.9 


Fi^.ll. 


J'-AJrVARV  /4,  /e3!^ 


For  notice  of  copyright,  see  page 


Tiediately  following  the  title  page. 


jrO^^  37yTITJ<.  CLA^^JV945e9. 


§  13  GEOMETRICAL   DRAWING  69 

the  curve  of  intersection  is  always  a  straight  line,  no  matter 
what  angle  the  two  axes  make  with  each  other. 

Fig.  3  shows  a  symmetrical  three-jointed  ell)o^v  formed 
by  the  intersection  of  three  cylindrical  surfaces.  The  diam- 
eter of  each  of  the  three  surfaces  is  1^".  The  center  lines 
of  the  surfaces  RAGS  and  M  N  P  H  z.x&  to  be  at  right 
angles  to  each  other;  then,  in  order  that  the  arrangement 
shall  be  symmetrical,  the  center  line  of  the  third  surface 
A  M H G  must  make  an  angle  of  45°  with  the  center  lines 
of  the  other  two. 

To  construct  the  elevation  as  shoAvn  in  the  figure,  draw 
the  two  center  lines  ;// //  and  /^  at  right  angles  to  each 
other;  they  intersect  at  6.  Lay  off  6'/=  1|-"  and  draw  an 
indefinite  line  RS  through  /  perpendicular  to  mii.  Make 
I R  equal  to  /5  =  1^  x  J  =  f",  and  draw  RA  and  5  G  par- 
allel to  nin.  Draw  (9  i'T  parallel  to  vin  and  1|^"  below  it. 
Through  the  point  (7,  where  R  S  and  O  K  intersect,  draw 
O  T  passing  through  6\  and  bisect  the  angle  RO  T  hy  the 
line  O  A^  which  intersects  RA  Siud  •  S  G  \n  A  and  G.  Lay 
oK  6  /  =  2^"  and  draw  PJ N  perpendicular  to  pq.  Make 
/P  =  JN=  H  Xi  =  f",  and  draw  PH  and  NM  parallel 
to  pq.  Draw  OAT  so  as  to  bisect  the  angle  T O  K\ 
O  M  intersects  P // and  N  M  in  //and  J/.  Finally,  draw 
AM2in&  GH. 

Fig.  4  shows  the  intersection  of  two  unequal  cylin- 
drical surfaces  whose  axes  intersect  at  an  angle  of  65° 
instead  of  90°,  as  in  Fig.  1.  The  method  of  finding  the 
curve  of  intersection  is  in  all  respects  similar  to  that  used 
in  Fig.  1,  and,  as  the  corresponding  points  have  been  given 
the  same  letters  or  figures,  the  directions  given  for  Fig.  1 
can  be  applied  to  Fig.  4  also. 

Fig.  5  shows  a  cylindrical  piece  of  iron  2f|"  in  diam- 
eter that  has  been  gradually  turned  down  to  1^^"  diameter, 
and  then  having  the  larger  part  flattened  on  two  sides. 
The  large  and  small  parts  of  the  piece  are  connected  by  a 
graceful  curve.  The  problem  is  to  find  the  curve  of  inter- 
section A  123 B  formed  by  the  flattening.  Draw  the  plan 
and  front  elevation  from  the  dimensions  given;  also  draw 


70  GEOMETRICAL   DRAWING  §13 

the  curve  C  6'  5'  Jf.\  and  its  equal  on  the  opposite  side,  sc 
that  they  look  to  the  eye  about  as  seen  in  the  drawing.  In 
order  that  all  the  work  sent  to  us  may  be  alike,  the  radius 
of  this  curve  and  the  position  of  the  center  have  been  given 
on  the  drawing.  To  locate  the  center,  draw  an  indefinite 
horizontal  straight  line  1"  -\-  Ijg"  =  2^"  above  the  base  of 
the  piece;  and  with  C ^nd  D  as  centers,  and  a  radius  of  \^' , 
describe  short  arcs  cutting  the  line  just  drawn.  The  points 
of  intersection  will  be  the  required  centers.  With  (9  as  a 
center,  and  radii  of  convenient  lengths,  as  (9^,  0  5,  0  6,  etc., 
describe  arcs  cutting  A'  B'  in  3' ,  2' ,  1' ,  etc.  Through  the 
points  ^,  5,  6,  etc.  draw  the  lines  4-4'>  3-5',  6-6',  etc.,  par- 
allel to  the  center  line  ///  n,  and  intersecting  the  curve  C  Jf!  in 
4',  5',  6',  C,  etc.  Through  the  points  A' ,  1' ,  2' ,  etc.  draw 
lines  A' A,  I'-l,  2'-2,  etc.,  parallel  to  nni,  intersecting  hori- 
zontal lines  drawn  through  C,  6',  5\  Jf! ,  etc.,  in  A,  1,  2,  3,  etc. 
The  points  A,  1,  2,  3,  etc.  are  points  on  the  required  curve, 
and  through  them  the  curve  may  be  drawn. 

Fig.  6  is  the  cylindrical  surface  of  one  section  of  the 
elbow  1  7  A  G  oi  Fig.  2  rolled  out  into  a  flat  plate ;  hence,  if 
a  flat  plate  were  cut  into  the  same  shape  and  size  as  Fig.  G 
and  bent  into  a  cylinder  so  that  the  ends  1  G'  and  I'G" 
touch  each  other,  the  vertical  projection  or  front  elevation 
would  be  the  same  as  shown  hy  1  7  A  G  in  Fig.  2.  If  a 
second  plate  were  cut  out  in  the  same  manner  and  bent  into 
a  circle,  the  two  pieces  on  being  brought  together,  as  shown 
in  Fig.  2,  would  touch  at  every  point.  The  problem  is  to 
find  the  shape  of  the  curve  G'  A'  G".  The  length  of  the 
line  1-T  is  evidently  equal  to  the  length  of  the  circumfer- 
ence of  a  circle  whose  diameter  is  If",  or  4.32",  very  nearly. 
Produce  the  line  1-7,  Fig.  2,  and  make  1-V  equal  in  length 
to  4.32".  Divide  the  circle  12 3....  12  into  a  convenient 
number  of  equal  parts,  in  this  case  12,  and  erect  the  per- 
pendiculars 1  G,  2 F,  3 E,  etc.,  cutting  the  line  of  intersec- 
tion G  A  oi  the  cylindrical  surfaces  in  G,  F,  E,  etc.  Divide 
the  line  1-1'  into  the  same  number  of  equal  parts  that  the 
circle  was  divided  into,  thus  making  the  length  1-2  equal 
length  of  arc  1-2;  2-3,  length  of  arc  2-3,  etc.     Through  i, 


§  13  GEOMETRICAL   DRAWING  71 

2,  3,  etc.,  draw  the  perpendiculars  1  G\  2  F\  3  E\  etc.  and 
project  the  points  G,  F,  E,  etc.  upon  these  perpendiculars, 
as  shown,  thus  locating  the  points  G' ,  F\  E' ,  D\  C,  B\  A' 
of  the  left-hand  half  of  the  required  curve.  The  points  on 
the  right-hand  half  are  found  in  the  same  manner,  as  shown, 
and  the  required  curve  can  be  drawn  through  these  points. 

50.     A  drawing  like  Fig.  G  is  called  the  development 

of  the  cylindrical  surface  17  A  G. 

Fig.  7  is  the  development  of  the  cylindrical  surface 

A  G  H  M  of  Fig.  3.  Make  1-1'=  Ux  3.1416  =  4.71", 
neaily,  and  divide  it  into  12  equal  parts  to  correspond  with 
the  12  equal  parts  into  which  the  dotted  circle  is  divided. 
Project  the  points  6,  5,  etc.  of  the  dotted  circle  upon  O  A  as 
shown,  thus  locating  the  points  B,  C,  etc.  Through  B,  C, 
etc.,  draw  B 6,  C5,  etc.,  perpendicular  to  O  T.  Make  1  G' 
=  1  G'"  =  1  G,2  F  =2  F'"  =  2  F,  3  E'  =  3  E'"  =  3  E,  etc. 
Through  G' ,  F',  E' ,  etc.,  trace  the  curve  G'  F'  E' .  .  .  .  G" ,  and, 
throug^h  G"\  F'".  E'",  etc.,  trace  the  curve  G'"  F'" E'" .  .  .  . 
C^'.      Drawing  G'  G'"  and  G"  G^""'  completes  the  figure. 

Fig.  8  is  the  development  of  the  cylindrical  surface  1 F E  A, 
Fig.  1.  The  method  used  here  is  in  all  respects  similar  to 
the  tAvo  preceding  problems.  In  this  case,  the  distances 
1  A,  7  E,  and  1'  A'  are  all  equal  to  lA  or  E F,  in  Fig.  1; 
and  2  B,  6  Q,  8  Q' ,  and  12  B'  are  all  equal  to  2  B  ov  6  Q,  in 
Fig.  1.  The  development  of  L  M P N"\s  not  given,  for  want 
of  room,  but  the  method  will  be  explained  in  Fig.  10. 

Fig.  9  is  the  development  of  the  cylindrical  surface 
1 FEA,  Fig.  4.  The  student  should  have  no  difficulty  in 
drawing  this,  after  having  studied  the  preceding  problems. 

Fig.  10  is  the  development  of  the  cylindrical  surface 
LMPN,  Fig.  4.  Owing  to  the  want  of  room,  only  that 
half  of  the  development  is  shown  which  contains  the  part  to 
be  cut  out.  The  length  of  a  circle  ly\"  in  diameter  is  4.9", 
nearly;  half  of  this  is  2.45".  Hence,  the  line  Y'Y", 
Fig.  10,  which  equals  the  length  of.  the  semicircle  V'A'Y", 
Fig.  4,  is  2.45"  long.  The  distance  X' V  =  X"  V"  equals 
the  length  of  the  cylinder,  L  X or  MP.     Lay  off  X'S  equal 


72  GEOMETRICAL   DRAWING  §  13 

to  the  length  of- the  arc  V U  \  5  Tv  equal  to  the  arc  D' C  \ 
i?  A' equal  to  the  arc  C B' \  A^ J/ equal  to  the  arc  B' A' ,  etc. 
Find  the  lengths  of  these  arcs  by  means  of  the  method 
given  in  connection  with  Fig.  19.  Draw  through  these 
points  the  perpendiculars  SS',RR',  etc.  With  the  spa- 
cing dividers,  set  off  5Z>,  equal  to  S /?  in  Fig.  4;  i?G^, 
equal  to  R  G;  X  Q^  equal  to  X  Q;  and  ME^  equal  to  ME. 
Also,  R'C^  equal  to  R'C;  N'B^  equal  to  N'B;  and  M'A^ 
equal  to  PA.  In  exactly  the  same  manner,  find  the  points 
on  the  right-hand  half  of  the  curve.  If  a  plate  were  cut 
of  the  same  size  and  shape  as  shown  in  Fig.  10,  and 
rolled  into  a  semicylindrical  surface,  the  diameter  of  which 
is  liV")  it  would  exactly  fit  the  plate  cut  like  Fig.  9  rolled 
into  a  cylindrical  surface,  the  diameter  of  which  is  1^",  the 
two  being  placed  together  as  shown  in  Fig.  4. 

Fig.  11  shows  a  conical  surface  cut  t>y  a  plane,  and 
Fig.  12  shows  its  development.  Draw  the  elevation  and 
horizontal  projection  of  the  base  as  shown  in  Fig.  11. 
Divide  the  projected  circle  (base  of  cone)  into  a  convenient 
number  of  equal  parts,  in  this  case  12,  and  project  the 
points  i,  2,  3,  etc.  on  the  base  l'-7',  thus  locating  the 
points  i',  2' ,  3\  etc.  Join  these  points  with  the  apex  O  of 
the  cone,  by  the  lines  0 1' ,  02',  03\  etc.,  cutting  the 
plane  in  A^  B,  C,  etc.  Now,  choose  a  convenient  point  O, 
Fig.  12,  and  with  this  as  a  center,  and  a  radius  equal 
to  0 1\  or  0  7',  Fig.  11,  the  slant  height  of  the  cone, 
describe  an  arc  1-1'  of  a  circle.  Make  the  length  of  this 
arc  equal  to  the  length  of  the  circumference  of  a  circle 
having  the  same  diameter  as  the  base  of  the  cone.  This 
may  be  conveniently  done  as  follows :  length  of  arc  =  2 
X  3.1416  =  6.28",  nearly.  Draw  a  straight  line  6.28"  long 
and  divide  it  into,  say,  4  equal  parts.  Describe  an  arc  hav- 
ing a  radius  equal  to  O  T ,  the  slant  height  of  the  cone,  and 
find  the  length  of  a  part  of  this  arc  equal  to  6.28  -f-  4 
=  1.57"  by  means  of  the  method  described  in  connection 
with  Fig.  48.  With  the  dividers  set  for  the  chord  of  the 
arc  just  found,  space  off  the  chord  four  times  on  the  longer 
arc   12  3  ....1',   Fig.   12.      Divide  the    arc    into    the   same 


§  13  GEOMETRICAL  DRAWING  73 

niiniber  of  equal  parts  tliat  the  circle  1  2  3.  .  .  .12  has  been 
divided  into,  that  is,  12  parts.  Join  the  points  of  divi- 
sion i,  2,  3,  etc.  with  the  center  O  by  the  lines  O  1,02,  O  3, 
etc.,  as  shown.  Project  the  points  B,  C,  D,  etc.,  Fig.  11, 
upon  O  1',  in  i),,  C,,  D^,  etc.,  as  shown,  and  lay  ofi  O  A  equal 
to  O  A'  equal  to  OA,  Fig.  11;  O  B  equal  to  O  B'  equal  to 
(9/>', ;  O  C  equal  to  O  C  equal  to  O  C^,  etc.,  and  through 
these  points  draw  the  curve.  A  plate  cut  of  the  same  size 
and  shape  as  shown  by  A  G  A'  T  7  1  can  be  bent  into  the 
conical  surface  shown  by  the  elevation  A  G  7'  1' . 

Particular  attention  must  be  given  to  the  method  explained 
above  for  laying  out  the  curve  of  t lie  development  in  Fig.  12. 
It  would  be  entirely  wrong  to  take  the  measurements  from 
the  lines  O  F,  O  E,  O  D,  O  C,  etc.,  Fig.  11.  The  reason  for 
this  is  that  these  lines,  being  on  the  surface  of  the  cone,  are 
inclined  tow^ards  the  observer,  and  so  do  not  appear  in  their 
true  lengths.  The  line  O  D,  for  example,  if  measured  on  the 
surface  of  the  cone  itself,  would  evidently  be  of  the  same 
length  as  the  line  O  D^\  but  in  the  figure  it  is  much  shorter. 
The  line  O D^,  however,  appears  in  its  true  length  in  the  figure, 
because  it  is  not  inclined  to  the  observer  in  the  position  shown. 
The  actual  distance  of  point  D  from  the  apex  (9,  therefore,  is 
O D^,  which  is  the  distance  to  be  laid  ofif  for  point  D  in  the 
development.     The  same  holds  true  for  the  other  points. 


SHADE  lilJ^ES 

51.  The  use  of  the  heavy  shade  line  will  now  be  ex- 
plained In  Fig.  71,  by  means  of  the  shade  lines,  the  drafts- 
man knows,  without  looking  at  any  other  view  of  the  object, 
that  the  rectangles  1  and  Jf.  represent  square  holes,  and 
2  and  3,  square  bosses.  When  he  looks  at  the  other  view, 
it  is  to  find  the  depth  of  the  holes  and  the  height  of  the 
bosses.  This  explains  the  use  of  the  shade  lines,  viz.  :  to 
show,  from  that  view  of  the  drawing  which  is  being 
examined,  whether  the  part  looked  at  is  above  or  below 
the    plane  of    the    surface:    that   is,    for  example,    whether 


74 


GEOMETRICAL   DRAWING 


13 


J z 

i  L>— — 

3  4: 


Fig.  71 


the    rectangles    1,    2,    3,    and  4-   are   the   tops   of    bosses    or 
bottoms  of   holes,   and,   consequently,  whether  they  extend 

above  or  below  the  sur- 
face  of  A  B DC.  In 
order  that  the  shading 
may  be  uniform  on  all 
drawings,  the  light  is 
assumed  to  come  in 
one  invariable  direc- 
tion, in  such  a  man- 
ner as  to  be  parallel 
to  the  plane  of  the 
paper,  to  make  an 
angle  of  45"^  with  all 
horizontal  and  verti- 
cal lines  of  the  draw- 
ing, and  to  come  from 
the  upper  left-hand  corner  of  the  drawing.  Each  view 
of  the  object  represented  is  shaded  independently  of  any 
of  the  others;  and,  when  shading,  the  object  is  always 
supposed  to  stand  in  such  a  position  that  the  drawing 
will  represent  a  top  view.  Any  surface  that  can  be  touched 
by  drawing  a  series  of  parallel  straight  lines,  making 
an  angle  of  45°  with  the  horizontal  and  vertical  lines  of 
the  drawing,  is  called  a  light  surface ;  a  surface  that 
cannot  be  touched  by  lines  having  this  angle  is  called  a 
(lark  surface.  All  of  the  edges  caused  by  the  inter- 
section of  a  light  and  dark  surface,  or  two  dark  surfaces, 
are  usually  shaded;  that  is,  the  edges  thus  formed  are 
drawn  in  heavy  lines.  Exceptions  to  this  rule  are  some- 
times made  by  experienced  draftsmen,  when  a  rigid  adher- 
ence to  it  will  produce  a  bad  effect  or  will  render  the 
drawing  ambiguous. 

Fig.  72  shows  a  plan  of  a  series  of  triangular  wedges  radi- 
ating from  the  common  center  O.  The  top  is,  of  course,  a 
light  surface,  and,  in  order  to  determine  whether  the  per- 
pendicular surfaces  are  light  or  not.  the  45°  triangle  maybe 
used.     Take  the  wedge  R  O  A.     A  line  drawn  at  an  angle  of 


§13 


GEOMETRICAL   DRAWING 


75 


45°,  the  direction  of  the  arrows,  would  strike  the  side  of 
which  0^4  is  the  edge;  hence,  this  side  is  a  Hght  surface, 
and  tlie  toi)  being  also 
a  light  surface,  the 
line  0.1  must  be  light. 
OK,  on  the  contrary,  is 
a  heavy  line,  since  the 
light  cannot  strike  the 
side  of  which  OK  is  the 
edge  without  passing 
through  the  wedge. 
Hence,  this  is  a  dark 
surface,  and  its  inter- 
section OK  with  the 
light  surface  OA  K  re- 
quires a  shaded  line. 
For  the  same  reason, 
AK    is    also    shaded. 

„,  .  FIG.  72 

1  he  same  reasonmg  as 

the  above  applies  to  the  lines  OB,  O D,  OG,  01,  OK, 
and  O  M\  also,  to  QN,  ML,  and  KJ.  C B  \s  not  shaded, 
because  the  light  strikes  the  surface  of  which  C  B  is  the 
edge,  as  shown  by  the  arrow,  making  C B  the  intersection 
of  two  light  surfaces.  (9  A^  makes  an  angle  of  exactly  45° 
with  the  horizontal,  and  is  treated  as  if  it  were  the  edge  of 
a  light  surface;  this  is  done  in  every  case  in  which  the  line 
considered  makes  an  angle  of  45°  with  the  horizontal. 

In  shading  holes,  or  any  parts  of  the  drawing  denoting 
depressions  below  the  surface  under  consideration,  a  slightly 
different  assumption  is  made.  Fig.  73  shows  the  plan  of  a 
square  block  with  a  hexagonal  hole  in  the  center.  If  the 
light  passed  over  the  surface  A  BCD,  parallel  to  the  plane 
of  the  paper  as  previously  assumed,  all  the  inside  sur- 
faces would  be  dark,  and  the  entire  outline  of  the  hexagon 
EFG H I K  would  be  shaded.  In  order  to  prevent  this 
and  make  the  work  similar  to  that  which  has  preceded, 
the  rays  of  light  are  assumed  to  make  an  angle  of  45° 
with    the    plane    of    the    paper    when    shading    holes    and 


76 


GEOMETRICAL   DRAWING 


13 


FIG.  73 


depressions.  Hence,  the  light  will  strike  the  surfaces  whose 
edges  are  G H^  HI,  and  /A",  as  shown  by  the  arrows,  leav- 
ing the  surfaces  whose  edges 
are  K E^  E F,  and  EG  dark  as 
before.  Therefore,  these  latter 
edges  will  be  shaded,  and  the 
edges  G  H,  HE  and  /A"  will  be 
light.      See  also  Fig.  71. 

The  conventional  method  of 
shading  circles  which  represent 
the  projections  of  cylinders,  or 
circular  holes,  is  as  follows: 
A  B,  Fig.  74,  is  the  projection 
or  end  view  of  a  cylinder  having 
for  a  base  the  circular  area  A  B.  Draw  the  arrows  E  A  and 
E B^  making  angles  of  45'  with  the  horizontal  diameter, 
and  tangent  to  the  circle  at  A  and  B.  That  half  of  the 
circle  in  front  of  these  two  points  of  tangency  is  to  be  shaded, 
and,  in  order  to  make 
the  drawing  look  well,  the 
center  point  for  the  com- 
passes is  shifted  along  the 
line  C  H  parallel  to  EA 
and  E B  in  the  direction 
of  the  arrow  an  amount 
equal  to  the  thickness  of 
the  desired  line.  With 
the  same  radius  that  was 
used  to  describe  the  orig- 
inal circle,  describe  part 
of  another  circle,  being 
careful  not  to  run  over 
the  first  circle,  and  stop- 
ping when  the  two  lines  coincide.  The  directions  for  sha- 
ding a  hole  are  precisely  the  same  as  for  the  projection  of 
a  cylinder  base,  except  that  the  half  B  C  A  oi  the  circle  in 
Fig.  75  is  to  be  shaded,  the  center  being  shifted  as  before, 
but  in  the  opposite  direction,  as  shown  by  the  arrow. 


Fig.  74 


§13 


GEOMETRICAL  DRAWING 


77 


Vertical  projections  of  cylinders  are  shaded  as  shown 
in  the  front  elevation  of  Fig.  5,  Drawing  Plate,  title: 
Projections — I. 

After  studying  the  foregoing  concerning  shade  lines,  the 
student  should  be  able  to 
see  the  reason  for  the  using 
or  omitting  of  any  shade 
lines  on  the  drawings  in  the 
following  plates.  In  the  case 
of  an  object  like  the  hex- 
agonal prism  in  Fig.  G, 
Drawing  Plate,  title:  Pro- 
jections— I,  no  part  of  the 
upper  base  or  line  Se  is 
shaded,  although,  strictly 
speaking,  the  part  ^  ^  of  the 
line  should  be  shaded ;  but,  as  this  would  make  part  of  the 
straight  line  S c  heavy  and  the  greater  part  light,  the  whole 
line  is  drawn  light.  This  is  one  of  the  exceptions  previously 
mentioned. 


Fig. 


MECHANICAL    DRAWING 


Plew. 


lop 


I  Frohk  iHew. 
A 


ce:n^ter  liio]S 

1.  Fig.  1  represents  a  thick  wedge  having  a  cyHndrical 
hole  running  through  its  entire  length.  The  lines  inii^pq^ 
and  }- s  are  called  center 
lines.  Center  lines  are  usu- 
ally drawn  through  the  cen- 
ter of  anything  that  is  round, 
such  as  a  cylinder  or  a  cylin- 
drical hole.  In  the  case  of  a 
circle,  there  are  usually  two 
center  lines,  one  being  at 
right  angles  to  the  other,  as 
shown  in  view  />,  Fig.  1. 
By  drawing  two  center  lines 
through  a  circle  in  the  man- 
ner just  mentioned,  the  cen- 
ter of  the  circle  is  located  by 
their  intersection.  The  mere 
presence  of  these  lines  shows 
in  most  cases  that  part  of  the 
object  through  which  they  are  drawn  is  round.  It  is  very 
seldom  that  center  lines  appear  on  drawings  unless  they  are 
the  center  lines  of  cylindrical  surfaces.  They  may  some- 
times be  drawn  to  indicate  that  the  surface  is  a  regularly 
curved  surface,  such  as  would  be  formed  by  circular  arcs,  or 

■      g  14 

For  notice  of  copyright,  see  page  immediately  following  the  title  page. 


£'^t^^ew. 


Fig.  1 


2  MECHANICAL  DRAWING  §  14 

one  having  the  shape  of  an  ellipse.  In  very  rare  cases,  for 
some  special  reason,  a  line  that  corresponds  to  a  center  line 
may  be  drawn  for  some  particular  purpose,  but  such  a  line 
is  not,  in  the  strict  sense  of  the  word,  a  center  line. 

2,  Center  lines  are  an  extremely  important  feature  of  a 
drawing,  since  the  workman  is  guided  by  them  in  doing  the 
work  called  for  by  the  drawing.  For  instance,  suppose  that 
it  was  required  to  make  a  wedge  like  that  shown  in  Fig.  1, 
and  that  the  workman  was  given  a  piece  of  cast  iron  having 
approximately  the  shape  indicated  by  the  drawing.  Sup- 
pose, further,  that  it  was  necessary  to  have  the  hole  located 
exactly  as  shown  in  the  drawing  with  reference  to  the  sides 
of  the  wedge,  and  that  the  sides  and  ends  of  the  wedge 
were  all  to  be  "finished."  The  first  thing  that  the  work- 
man would  probably  do  would  be  to  drill  the  hole,  and,  if 
the  job  had  to  be  very  accurate,  he  would  drill  the  hole  a 
little  smaller  than  the  drawing  calls  for  and  then  ream  it 
out  to  size.  He  would  then  face  the  ends  square  with  the 
center  line  of  the  hole  and  make  the  length  of  the  wedge 
the  same  as  shown  on  the  drawing.  The  sides  of  the  wedge 
would  then  be  planed  and  finally  finished  with  a  file,  the 
workman  working  all  the  time  from  the  center  line  ;//;/. 
If  the  drawing  shown  in  Fig.  1  is  intended  to  be  worked  to, 
it  would,  in  most  shops,  be  supplied  with  proper  dimensions. 


SECTIOIS'S  A30>  SECTIOX  LIXTN'G 

3.  In  order  to  show  the  interior  of  hollow  objects,  they 
are  often  drawn  in  section,  and  the  kind  of  material  is  then 
usually  indicated  by  certain  combinations  of  lines.  Unfor- 
tunately, there  is  no  universally  adopted  standard ;  thus,  a 
certain  combination  of  lines  may  indicate  that  the  material 
is  cast  iron  if  drawn  in  one  office;  in  another  office  this 
same  combination  may  have  been  adopted  to  represent 
brass,  and  so  on.  As  far  as  working  drawings  are  con- 
cerned, there  is  usually  no  difficulty  experienced  on  account 
of  this  diversity  of  practice,  since  as  a  general  rule  the 
material  is,  and  should  always  be,  distinctly  specified  on  the 


§14 


MECHANICAL  DRAWING 


drawing   in   order  to  prevent   any   mistake  on  the  part  of 
the  workman. 

4.     The   most   commonly  used  combination   of    h'nes  for 
different  materials  is  shown  in  Fig.  2.      Steel  of  all  kinds  is 


^^y'yyWyS'yZ. 


'WM^/M\      f;V^""'""V'-'\     f--— ^^     ^^mrrmmmvij,. 


D 


j: 


G 


m. 


H 


Fig.  2 


indicated  as  shown  in  view  A  ;  view  B  shows  the  style  of  sec- 
tioning employed  for  wrought  iron.  Cast  iron  is  usually  sec- 
tioned as  shown  at  C\  brass  and  other  similar  copper  alloys 
are  sectioned  in  the  manner  shown  at  D.  For  lead,  Babbitt, 
and  similar  soft  metal,  the  sectioning  shown  at  E  is  exten- 
sively used.  Wood,  when  cut  across  the  grain,  is  usually  sec- 
tioned as  shoAvn  in  the  upper  half  of  view  F^  and  Avhen  cut 
along  the  grain,  as  shown  in  the  lower  half.  Wood  is  also- 
frequently  indicated  on  a  drawing  by  section  lines,  even  when 
it  is  not  a  section.  Glass  and  stone,  when  in  section,  are 
often  indicated  in  the  manner  shown  by  the  upper  half  of 
view  G\  when  not  in  section,  they  are  frequently  drawn  as 
shown  in  the  lower  half  of  that  view.  Concrete  may  be 
indicated  as  in  view  H\  view  /gives  a  common  representa- 
tion of  leather.  Rubber  and  wood  fiber  are  sectioned  as  in 
viewy,  firebrick  as  in  /v",  and  water  as  in  L. 


5.     Instead  of    representing  sections  by   lines,   they  are 
occasionally  colored,  the  colors  used  indicating  the  different 
materials.      While  this  practice  is  very  common  in  Europe, 
it  is  very  rarely  found  in  the  United  States. 
M.  E.     v.— 7 


MECHANICAL  DRAWING 


14 


6.     Sections  of  material  that  appear  too  thin  on  a  draw- 
ing to  be  conveniently  sectioned,   or  when  it  is  desired  to 

make  the  section 
very  prominent,  are 
often  blackened  in, 
as  shown  in  Fig.  3. 
In  order  to  separate 
■i^H^B  different  pieces,  a 
Fig.  3  white    line    is   then 

usually  left  between  them.  Black  sections  are  most  fre- 
quently employed  for  sectional  views  of  structures  com- 
posed of  plates  and  rolled  sections,  such  as  I  beams,  angle 
irons,  bulb  angles,  rails,  Z  bars. 


Fig.  4 


7.  On  many  sectional  views,  it  will  be  noticed  that  the 
section  lines  do  not  run  in  the  same  direction.  This  invari- 
ably means  that  there  is  more 
than  one  piece  in  the  section 
given.  Thus,  referring  to  Fig.  4, 
it  will  be  seen  that  the  section 
lining  shown  at  b,  b  is  at  a  right 
angle  to  the  other  section  lining. 
It  is  the  general  rule  among 
draftsmen  that  all  parts  of  the  same  piece  shown  in  section 
must  be  section-lined  in  the  same  direction,  irrespective  of 
the  continuity  of  the  section.  Thus,  referring  again  to 
Fig.  4,  the  fact  that  all  section  lining  marked  A  is  in  the 
same  direction  immediately  establishes  the  fact  that  this 
part  of  the  view  is  a  section  of  the  same  piece.  Likewise, 
since  the  sectioning  shown  at  ^,  b  runs  in  the  same  direc- 
tion, it  follows  that  b,  b  are  sectional  views  of  one  piece, 
which  is  separate  from  A. 

8.  The  above  rule  governing  the  direction  of  section 
lines  is  always  adhered  to  when  possible ;  when  any  depar- 
ture is  necessary,  care  is  taken  to  prevent  ambiguity.  Where 
only  the  sectional  view  is  given,  it  is  often  very  difficult  to 
understand  the  drawing,  and  sometimes  a  violation  of  the 


S  U 


MECHANICAL  DRAWING 


above  rule  will  cause  an  erroneous  conclusion  to  be  drawn. 
Referring  to  Fig.  5  (a),  cover  up  the  front  view  shown  at  {/?). 
Then,  since  the  sectioning  of  A  and  B,  and  also  that  shown 


(a) 


Rocis    removed, 
(b)  . 


at  b  and  b' ,  are  respectively  in  the  same  direction,  any  one 
would  be  perfectly  justified  in  assuming  that  .  /  and  B  was 
a   sectional    view   of  a  rod    fitted   with   a   solid    bushing  b. 


6  MECHANICAL  DRAWING  §  U 

Furthermore,  since  C^  C  and  C\  C  are  sectioned  the  same 
way,  the  conclusion  that  they  were  the  jaws  of  a  forked 
rod  would  be  justifiable.  Referring  now  to  view  {b),  it  is 
seen  that  b  and  b'  are  separate  brass  boxes;  the  part  B 
is  seen  to  be  separate  from  the  cap  A,  and  the  note 
'^  Rods  removed'"  indicates  that  C  is  separate  from  C\ 
The  way  the  sectional  view  should  have  been  section-lined 
to  correspond  to  the  front  \"iew  shown  at  {b)  is  given  in 
Fig.   5  if). 

9.  When  a  cutting  plane  passes  through  the  axis  of  a 
shaft,  bolt,  rod,  or  any  other  sohd  piece  having  a  curved 
surface  and  located  in  the  plane  on  which  the  section  is 
taken,  it  is  the  general  practice  not  to  show  such  solid  pieces 
in  section,  but  in  fuU.  Thus,  in  Fig.  5  the  sectional  view  is 
taken  on  the  plane  represented  by  the  line  x  y^  which  passes 
through  the  axis  of  the  pin  D.  This  pin  is  shown  in  full, 
however.  The  practice  here  shown  is  rarely  departed  from 
by  experienced  draftsmen,  since  it  makes  a  drawing  easier 
to  read  and  also  saves  considerable  time  in  making  the 
drawing. 

10.  Fig.  5  also  shows  another  feature  that  is  frequently 
met  with  in  shop  drawings.  Referring  to  the  illustration, 
it  is  seen  that  no  bolt  is  shown  in  the  lower  half  of  the 
object,  as  far  as  the  front  view  {b)  is  concerned.  A  center 
line  op  is  drawn  in,  however ;  this  center  line  indicates  to 
the  workman,  who  reasons  from  the  symmetry  of  the  object 
in  respect  to  the  center  line  x }\  that  the  lower  half  of  the 
object  is  to  be  supplied  with  a  bolt  placed  in  the  plane 
given  by  the  center  line  op.  In  case  of  symmetrical  work, 
draftsmen  wiU  frequently  complete  only  one  half  of  the  view 
and  merely  indicate  the  other  half  by  a  few  lines  or  not  at 
all,  trusting  to  the  judgment  of  the  workman  for  a  correct 
reading  of  the  drawing.  In  the  best  practice,  a  note  is 
made  on  the  drawing  calling  attention  to  the  fact  that 
the  indicated  portion  of  the  A-iew  is  a  duplicate  of  the 
complete  portion. 


§  li 


MECHANICAL  DRAWING 


BREAKS 

11.  When  a  long-  and  comparatively  slender  object  is  to 
be  drawn,  it  often  happens  that,  when  drawn  to  a  sufBciently 
large  scale  to  make  it  intelligible,  it 
will  extend  beyond  the  space  avail- 
able. In  such  a  case,  part  of  the 
object  is  broken  out  and  the  remain- 
ing ends  are  pushed  together.  The 
fact  that  part  of  the  object  is  broken 
away  for  the  sake  of  convenience  is 
indicated  by  a  so-called  break.  It 
is  always  understood  that  the  part 
broken  away  and  not  shown  is  of  the 
same  size  and  shape  as  the  parts  con- 
tiguous to  the  break.      In  some  cases, 

one  end  of  the  object  is  broken  awav 

(e)  ^ 


(b) 


(c) 


(d) 


(f) 


(g) 


(It) 


Fig.  6 


] 


Z2 
ID 

U 


1'^.     Breaks    may   be    indicated    in 
various    ways;    most    commonly,    the 
break    is  given    an    outline    that  will 
reveal  the  shape  of  the  object.     Con- 
ventional methods  of  indicating  breaks 
are  shown  in  Fig.  G.     Wood  is  usually 
shown    broken    in    the    manner    illus- 
trated   at   {a),   angle   irons  as  at  {b), 
T  irons  as  at   (r),   Z  bars  as  at   {d). 
Cylindrical    objects    are    occasionally 
broken    as    shown    at   (r),    but    most 
frequently     in     the     manner     shown 
'^t    {/)■       Pipes    and    similar    hollow 
cylindrical  objects  may  be  broken  as 
shown   at   (,;') ;   but,  more  frequently, 
the   break  is  made  as   shown   at    (//). 
Rectangular  objects    may   be    broken 
in  the  manner  shown   at    (z);    plates 
and  objects  other  than  those  included 
between  views  (c?)  and  (/)  are  often  shown  broken  off  by 
drawing  a  wavy  freehand  line  as  in  (/)  and  {k). 


1 


MECHANICAL  DRAWING 


?  1-t 


HIDDEX    SCREW    THREADS 

13.  When  the  screw  thread  is  hidden  by  part  of  the 
object  and  it  is  deemed  necessary  to  show  it  in  dotted  lines, 
it  is  usually  drawn  in  one  of  the  four  ways  illustrated  in 
Fig.  7.     Of  these  the  method  shown  in  Fig.  T  (a)  is  probably 


r 


V - 


Fig.  7 

the  clearest;  that  shown  in  Fig.  7  (d)  is  fairly  good;  and  the 
one  shown  in  Fig.  7  (c)  is  cheap.  The  method  illustrated 
in  Fig.  7  (</)  is  practically  no  representation  of  a  screw 
thread  at  all;  if  used  it  usually  must  be  supplemented  by  a 
note  such  as  "f  stjid"  or  "  1^'  bolt,'"  and  so  on. 


REPEATED    PARTS    OF    OBJECTS 

1-4.  When  an  object  has  a  relatively  large  number  of 
similar  component  parts,  they  are  rarely  all  shown  on  a  work- 
ing drawing.  Usually,  a  few  of  them  are  shown  in  full  and 
the  rest  are  merely  indicated  by  showing  the  position  of  the 
center  of  each  part.  Sometimes  even  this  is  not  done,  but  a 
note  is  placed  on  the  drawing  calling  attention  to  the  fact 
that  some  certain  part  of  the  object  is  to  be  repeated. 

15.  Fig.  8  is  an  example  of  how  repeated  parts  of  an 
object  may  be  treated.  Referring  to  the  illustration,  which 
is  a  top  view  of  a  pipe  flange,  Fig.  8  {a)  shows  three  bolts 
drawn  in.  The  position  of  the  rest  of  the  bolts  is  indicated 
by  the  short  radial  lines  drawn  across  the  bolt  circle.     In 


§l-t 


MECHANICAL  DRAWING 


9 


Fig.  8  (d),  the  bolt  circle  is  drawn  in  and  a  note  is  Avritten 
along  it  that  is  sufficiently  definite  to  convey  the  idea  to  the 


Fig.  8 
mind  of  the  workman.     This  latter  method  is  most  commonly 
used  on  working  drawings. 

16.  When  making  a  drawing  of  a  gear-wheel,  especially 
when  it  is  a  working  drawing,  it  is  customary  to  draw  in 
only  two  or  three  teeth  and  specify  how  many  teeth  the 
wheel  is  to  have.  This  answers  the  purpose  as  well  as  if  all 
the  teeth  were  drawn  and  has  the  advantage  that  an  enor- 
mous amount  of  time  is  saved  in  making  the  drawing.  Like- 
wise, in  drawings  of  objects  built  up  of  plates  or  rolled  sec- 
tions, as  in  boiler  and  bridge  drawings  and  similar  work, 
only  a  few  rivets,  or  staybolts,  or  similar  repeated  parts 
arc  usually  shown,  and  the  location  of  the  rest  is  indicated 
by  showing  the  position  of  their  centers  or  bv  a  suitable 
note  placed  on  the  drawing. 


ABBREVIATIONS    USED    OX    DRAWINGS 

17.  The  most  commonly  used  abbreviations  are  D., 
D/a.,  D/a/j/.,  (/.,  dia.,  and  (/ia///.  for  "diameter,"  and  A^, 
Rett/.,  ?■.,  and  rad.  for  "  radius."  *'  Wrought  iron  "  is  usually 
abbreviated  to  f[>V  /ro//.  The  abbreviation  77/(/s.  or  ///t/s., 
with  a  number  prefixed,  stands  for  "  threads  per  inch  "  ;  thus, 
14  thds.  means  "  make  14  threads  per  inch."     The  word  tap, 


10  MECHANICAL  DRAWING  §  14 

with  a  number  prefixed,  always  means  that  a  hole  is  to 
be  finished  by  tapping  it  with  a  tap  of  standard  proportions, 
having  the  diameter  given  by  the  prefixed  number.  When 
a  tap  other  than  a  standard  tap  is  to  be  used,  it  is  distinctly 
specified.  Drill  is  always  taken  to  mean  that  a  hole  is  to  be 
put  through  the  object  by  drilling.  Bored  or  Bore  means 
''finish  a  hole  by  boring  it."  Planed  is  always  understood 
to  mean  *'  this  surface  is  to  be  finished  by  planing."  Cored 
implies  that  the  hole  to  which  it  is  applied  is  to  be  cored  out 
and  left  that  Avay ;  that  is,  it  is  not  to  be  finished  by  machin- 
ing. Faced  almost  invariably  implies  that  the  surface  to 
which  it  is  applied  is  to  be  machined  square  with  a  hole  in 
the  object.  Turtied \s  an  abbreviation  for  ''finish  by  turn- 
ing." Scraped  implies  that  a  surface  is  to  be  finished  by 
scraping.  Tool fiuish  means  that  the  surface,  after  machin- 
ing, is  not  to  be  finished  any  further.  Black,  on  objects 
formed  by  forging,  implies  that  the  part  to  which  it  is 
applied  is  to  be  left  as  it  comes  from  the  smith.  The  term 
Ream  or  Reauied  means  that  a  hole  is  to  be  finished  by  ream- 
ing; when  applied  to  a  bolt,  it  is  understood  that  the  bolt  is 
to  be  fitted  to  a  hole  that  has  been  previously  reamed.  The 
terms  Shrinking  Fit,  Forcing  Fit,  and  Driving  Fit  written 
behind  a  dimension  always  imply  that,  in  machining  the 
part,  the  workman  is  to  make  the  allowance  necessary  for 
the  kind  of  fit  called  for.  The  fact  that  part  of  an  object  is 
to  be  finished  by  machining,  filing,  or  grinding  is  often 
indicated  by  marking  the  outlines  with  an  f  written  across 
or  near  it,  or  writing  yf;/.  along  it.  In  some  cases,  draftsmen 
will  draw  a  dotted  line  or  a  full  red  line  at  a  little  distance 
from  the  outline  and  write /"across  it;  it  is  usually  under- 
stood, in  that  case,  that  the  lengths  of  the  supplementary 
lines  denote  the  extent  of  the  surfaces  that  are  to  be  finished. 


KIXDS    OF   TTORKIXG    DKAWIXGS 

18.  Working  drawings  are  divided  into  two  general 
classes,  which  are:  assembly,  or  general,  drawings,  and  detail 
drawiiisrs. 


§  14  MECHANICAL  DRAWING  11 

Assembly,  or  j>-eiieral,  (ll•a^villy•s  show  the  workman 
the  relation  between,  and  the  places  or  positions  occupied 
by,  the  different  component  parts  of  a  structure,  machine, 
device,  fixture,  implement,  etc.  If  any  dimensions  are 
given,  they  are  usually  only  leading  dimensions. 

Detail  dra^vinjifs  show  the  exact  shape  and  size  of  each 
integral  part.  For  this  purpose  they  are  supplied  with  all 
the  dimensions  required  by  the  workman  and  any  additional 
explanatory  notes  that  the  draftsman  may  consider  nec- 
essary. 

Detail  drawings  may  be  made  so  complete  that  they  will 
answer  for  the  patternmaker,  blacksmith,  and  machinist, 
and  they  are  usually  so  made  in  the  smaller  shops.  In 
the  large  shops,  however,  separate  drawings  are  often 
made  for  the  patternmaker,  blacksmith,  and  machinist; 
the  detail  drawing  for  the  use  of  the  patternmaker,  then, 
contains  only  the  dimensions  and  notes  needed  by  him 
to  make  the  pattern;  that  for  the  blacksmith  contains 
the  dimensions  needed  for  making  the  forging;  and,  finally, 
that  for  the  machinist  contains  all  dimensions  needed 
by  him. 

19.  Attention  is  called  to  the  fact  that  practice  varies 
somewhat  in  different  places  in  regard  to  the  dimensions 
given  on  detail  drawings,  at  least  as  far  as  drawings  for  the 
patternmaker  and  blacksmith  are  concerned.  In  some 
places,  the  dimensions  given  represent  the  size  the  object  is 
to  be  when  Jill ishi'd ;  hence,  the  blacksmith  or  patternmaker 
must  make  necessary  finishing  allowances  himself.  In  other 
places,  again,  the  finishing  allowance  has  been,  and  usually  is, 
made  by  the  draftsman ;  the  dimensions  given  are  then  those 
of  the  pattern  or  forging.  If  in  doubt  about  the  practice 
followed  in  a  particular  drawing  office,  it  is  a  good  plan  to 
find  out  by  inquiry  what  system  is  used  in  the  shop  under 
consideration.  In  the  best  modern  practice,  a  note  calling 
attention  to  the  fact  that  the  sizes  given  are  those  when 
finished  is  placed  on  the  drawing;  thus,  "  All  finisJied sizes  " 
or  some  similar  note. 


12 


MECHAXICAL  DRAWING 


1-i 


\^ 


SCALES 

20.  When  it  is  desired  to  make  a  drawing 
other  than  full  size,  special  scales  are  used. 
Thus,  suppose  it  is  required  to  make  the  draw- 
ing \  size ;  then,  3  inches  on  the  drawing  would 
represent  1  foot  on  the  object.  Hence,  if 
3  inches  are  laid  off  and  divided  into  12  equal 
parts,  each  of  these  parts  will  represent  1  inch 
on  the  object.  If  these  parts  be  subdivided 
into  •2,  -i.  8,  etc.  parts,  each  will  represent 
3.  ^.  i.  etc.  of  1  inch  on  the  object.  A  scale 
of  this  kind  is  called  a  quarter  scale,  or  a 
scale  of  3  iuclies  to  tlie  foot.  An  eighth 
scale,  or  a  scale  of  1^  inclies  to  tlie  foot, 
would  be  constructed  in  the  same  way,  except 
that  \\  inches  would  be  laid  off  instead  of 
3  inches.  These  scales  are  written  3"  =  1  ft., 
U'  —  1  ft. 

21.  Fig.  9  shows  a  scale  which  is  con- 
venient for  the  student,  inasmuch  as  it  com- 
bines eleven  different  systems  of  subdivision 
and  may  be  used  for  all  the  work  ordinarily 
done  in  a  drafting  room.  This  scale  is  tri- 
angular in  section  and  13  inches  in  length, 
and  on  each  of  its  edges  is  laid  off  a  scale, 
as  shown  at  A^  B,  and  G.  The  scale  at  G 
is  "full  size";  that  is,  this  edge  of  the  scale 
is  divided  into  inches  and  fractions  of  an  inch 
down  to  sixteenths,  and  is  used  for  drawings  in 
which  an  object  is  represented  in  its  natural 
size.  On  its  opposite  side,  at  B,  is  shown 
the  quarter-sized  scale  of  3"  =  1  ft.  The  first 
3-inch  (actual  size)  di\nsion,  from  B  to  C,  is 
subdivided  into  12  parts  representing  inches, 
and  each  inch  is  then  divided  into  proportional 
fractions  of  an  inch,  generally  eighths.  From 
C  toD,  D  to  E,  and  E  to  F,  the  scale  is  marked 


§  U  MECHANICAL  DRAWING  13 

in  its  main  divisions  of  1  foot  each,  each  foot  being  3  inches 
long,  actual  size.  From  A  to  B  the  scale  is  independently 
divided  into  spaces  of  U  inches  (actual  size)  to  form 
an  eighth-sized  scale,  or  lf=l  ft.,  the  divisions  of  the 
latter  occurring  on  and  between  the  marks  for  the  3-inch 
scale. 

The  other  sides  and  edges  of  the  instrument  are  divided 
into  scales  of  1  inch  and  ^  inch,  f  inch  and  f  inch,  ^  inch 
and  ^  inch,  and  j\  inch  and  ^%  inch  to  the  foot.  Different 
makers  do  not  always  arrange  their  scales  in  the  same  man- 
ner. Thus,  instead  of  having  a  full-size  scale  and  scales  of 
3"  =  1  ft.  and  IV  =  1  ft.  on  one  side,  as  shown  in  Fig.  9, 
some  makers  have  the  full-size  scale  and  j\"  =  1  ft.  and 
-i/  =  1  ft.  on  one  side.  It  will  be  observed  that  the  num- 
bering of  the  feet  on  these  scales  does  not  start  at  the  end  of 
the  instrument,  but  at  the  first  division  from  the  end.  Thus, 
on  the  quarter-sized  scale  the  zero  mark  is  placed  at  C  and 
the  first  foot  is  measured  to  D.  This  is  done  so  that  the 
feet  and  inches  may  be  laid  off  independently  and  with  one 
reading  of  the  scale. 

The  figures  indicating  the  number  of  feet  on  this  scale 
are  placed  along  the  extreme  upper  edge  at  D,  £,  and  F, 
the  numbers  running  in  a  direction  away  from  the  part 
containing  the  inches.  The  numbers  indicating  inches 
run  in  an  opposite  direction  from  those  defining  the 
feet. 

To  lay  off  2  feet  3f  inches  on  a  scale  of  3"  ==  1  ft.  and 
from  a  given  point,  place  the  scale  on  the  point  so  that  the 
2-foot  mark  will  be  directly  over  it;  then  from  the  zero 
mark  C  lay  off  3J  inches,  as  shown,  locating  a  second 
point.  The  length  of  the  distance  thus  laid  off  between 
the  two  points  represents  2  feet  3J  inches.  The  scale  of 
1^"  =  1  ft.  is  used  in  a  similar  manner  to  lay  off  the 
same  distance.  The  figures  indicating  feet  on  this  scale 
are  placed  nearer  the  edge,  in  order  to  prevent  confusion 
in  reading. 

To  draw  to  half  size,  or  d  inches  to  the  foot,  use  the  full- 
size  scale,  and  remember  that  every  ^  inch  on  that  scale 


14  MECHAXICAL  DRAWING  §  14 

corresponds  to  1  inch  on  the  object,  that  is,  that  every 
dimension  is  only  half  of  the  real  length.  To  lay  oflF 
of  inches,  lay  oflF  5  half  inches  and  ^  of  an  inch  over; 
the  result  is  a  line  5f  inches  long  to  a  scale  of  6  inches 
to  the  foot. 

If  it  is  desired  to  draw  to  a  scale  of  f  of  an  inch  to  the 
foot,  or  -^^  size,  the  scale  of  1|^  inches  to  the  foot  may  be 
used  if  the  draftsman  has  no  scale  of  f  =  1  ft.,  halving  all 
dimensions,  as  in  the  previous  case  of  drawing  to  a  scale 
of  6  inches  to  the  foot  with  a  full-size  scale.  It  sometimes 
happens  that  a  draftsman  is  obliged  to  make  a  scale,  when 
the  size  of  his  plate  is  limited  and  a  general  drawing  of  some 
object  is  desired.  By  general  drawing  is  meant  a  complete 
view  of  the  object  in  plan  and  also  one  or  two  elevations. 
In  such  a  case,  one  scale  may  be  too  large  to  enable  the 
drawing  to  be  made  on  a  sheet  of  the  required  size ;  another 


^^ 


scale  may  make  it  too  small  to  show  up  weU.  For  example, 
a  \  scale  may  be  too  large  and  a  ^  scale  too  small ;  a 
^  scale  may  be  just  right.  If  the  draftsman  has  no  ^  scale 
(that  is,  a  scale  of  1  inch  to  the  foot),  he  may  make  one  by 
taking  a  piece  of  heavy  drawing  paper  and  cutting  out  a 
strip  about  the  size  of  an  ordinary  scale  and  laying  off  the 
inch  divisions  on  it.  Each  division  or  part  will  represent 
1  foot  on  the  object.  Divide  one  of  the  end  parts  into 
\%  equal  parts  and  each  will  represent  1  inch  on  the  object. 
Lines  indicating  half  and  quarter  inches  may  be  drawn  if 
considered  necessary. 

Fig.  10  shows  part  of  a  scale  made  in  this  manner,  giving 
feet,  inches,  and  half  inches — ^the  quarters,  eighths,  etc.  of 
an  inch  being  judged  by  the  eye. 


SU 


MECHANICAL  DRAWING 


15 


COXTENTIONAL.    REPRESENTATI0:N^    OF    A    NUT 

22,  Fig.  11  shows  the  ordinary  conventional  method  of 
representing  a  nut.  The  bottom  of  the  thread  is  1^  inches 
in  diameter  and  is  represented  by  the  dotted  circle;  this 
shows  that  it  is  intended  for  a  screw  li  inches  in  diameter. 
The  height  of  the  nut  equals  the  diameter  of  the  bolt  or 
screw  on  which  the  thread  is  cut.  The  two  views  on  the 
center  line  m  n  should  be  drawn  without  difficulty.  To 
draw  the  curves  e  a  and  a  d,  project    b  and  c  at  right  angles 


to  /  V  in  the  points  d,  a,  and  c\  pass  arcs  of  circles  through 
e  and  a  and  through  d  and  a  tangent  \.o  f  g,  finding  the  cen- 
ters of  these  arcs  by  trial.  The  best  way  of  doing  this  is  to 
draw  lines  parallel  to  /  v  midway  between  c  a  and  between  a 
and  d.  Then,  by  trial  with  the  compasses,  find  a  center  on 
these  lines  such  that  an  arc  struck  with  the  compasses  from 
this  center  will  pass  through  ^and  a  (or  a  and  d)  and  be  tan- 
gent to  f  g.  In  the  right-hand  view,  the  radius  of  the  arc  be 
is  the  same  as  the  height  of  the  nut;  the  centers  of  the  other 
two  arcs  are  found  by  trial  in  the  manner  just  described. 


16  MECHANICAL  DRAWING  §  14 

DRATTTNG    PI^VTZ.    TITLE:     DETAILS 

23.  The  first  eight  figures  of  this  plate  show  the  conven- 
tional methods  of  representing  screws.  The  actual  projec- 
tion of  a  screw  thread  will  be  similar  to  the  projection  of  a 
helix;  but  in  order  to  save  the  time  required  to  locate  the 
points  and  trace  in  the  curves,  the  following  methods  are 
universally  used,  except,  perhaps,  in  the  case  of  screws  of 
ven,-  large  diameter  and  pitch,  drawn  full  size. 

24.  Fig.  1  represents  a  single  square-tlii-eaded  screAv 
Ih^  inches  in  diameter  and  f  inch  pitch.  To  draw  the  screw, 
first  draw  the  center  line  ///  fi  and  a  line  A  B  at  right  angles 
to  it.  Make  the  distance  A  B  equal  to  the  diameter  of 
the  screw,  or  \\  inches,  and  through  the  points  A  and  B 
draw  lines  AD  and  BE  parallel  to  the  center  line  ;//;/. 
Also  lay  off  on  the  line  A  B  distances  A  F  and  B  G  equal  to 
one-half  of  the  pitch,  and  through  the  points  F  and  G  draw 
lines  FH  and  GI  parallel  to  the  center  line  ;//;/.  These 
lines  show  the  depth  of  the  thread.  On  the  line  A  D  lay 
off  the  width  of  the  thread  and  of  the  groove,  A  C,  C J^ 
J K,  etc.,  each  equal  to  one-half  of  the  pitch,  or  f  x  i  = 
^  inch.  Draw  the  line  BC^  and  through  the  points  y,  K^ 
L,  J/,  etc.,  draw  lines  parallel  to  BC.  Draw  faint  pencil 
lines  through  the  points  C  and  P,  J  and  Q^  K  and  R^  etc.    to 

represent  the  back  edges  of  the  threads,  and 
::ake  the  parts  that  are  seen  full  lines;  then 
:raw  the  lines  T  V,  UW,  etc.  The  method  of 
drawing  the  remainder  of  the  screw  and  the 
reason  for  using  the  heav\'  shade  lines,  as  shown, 
should  be  apparent  without  further  explanation. 
It  will  be  noticed  that  the  width  of  the  thread 
and  of  the  groove,  measured  parallel  to  the  cen- 
ter line  ///«,  and  the  depth  of  the  thread  are  all 
exactly  the  same ;  that  is,  the\-  are  each  equal  to 
one-half  of  the  pitch.  If  a  section  were  taken 
through  the  center  line  in ;/,  the  thread  and 
groove    would   look   like    Fig.   1-2,   a    series   of 


S^;. 


Fig.  1-2        squares,  hence  the  term  square  thread. 


DET/ 


JUNE  25.  /693. 


For  notice  of  copyright,  see  page  it 


L5. 


'■esentzrz.Q  Screu/3 


m_ 


T 


± 


CBAN/6. 

Wrnuaht  Iron^ 

Fu.llST.ee 
Fintsheai 
Ft,y  /O. 


diately  following  the  title  page. 


JOHN  SMITH.  CLASS  N9  4-529. 


§  14  MECHANICAL  DRAWING  17 

25.     Fig.    2    sliows    a    double    square-threaded    screw 

1^  inches  in  diameter  and   with  f  of  an  inch  pitch.     The 
reason  for  using  a  double  thread  is  that  if  the  single  square 
thread  were  used,  the  depth  would  be  so  great  as  to  weaken 
the  bolt  or  rod  on  which  it  was  cut  and  render  it  unsafe  for  the 
purpose  for  which  it  was  intended.     To  prevent  this,  either 
the  diameter  of  the  rod  must  be  increased  or  the  thread  must 
be  cut  of  the  same  depth  and  thickness  as  a  thread  of  half 
the  pitch,  or,  in  this  case,  as  if  the  pitch  were  f  X  i  ==  f  of 
an  inch,  as  in  the  preceding  problem;  another  thread  of  the 
same  size  and  pitch  (J  of  an  inch)  must  be  cut  half  way 
between  these  first  threads,  thus   giving  a  double   thread. 
The  pitch,  or  distance  that  the  screw  would  advance  in  one 
turn,  would  be  f  of  an  inch,  the  same  as  if  it  were  a  single- 
threaded  screw  of  f  of  an  inch  pitch,  while  the  depth  of  "the 
thread  is  only  half  as  great.     To  draw  it,  proceed  exactly  as 
in  the   last  figure.     To  get   the  direction  of  the  line  B  C, 
which  in  this  figure  represents  the  projection  of  the  bottom 
edge  of  the  top  of  the  thread,  lay  off  A  C  equal  to  one-half 
of  the  pitch,  or  f  x  i  =  f  of  an  inch,  and  draw  the  line  B  C. 
The  width  of  the  threads  and  grooves,  and  also  the  depth  of 
the  threads,  is  one-fourth  of  the  pitch,  or  f  x  i  =  yV  inch. 
Through  the  points  A',  Z,  J/,  etc.  draw  faint  pencil  lines  A" A^ 
etc.  to  represent  the  back  edges  of  the  threads,  and  make  the 
parts  that  are  seen  full  lines.     Through  the  point  A' draw  a 
faint  pencil  line  A' A'  at  right  angles  to  the  center  line  vni, 
intersecting  the  line  F H  in  a,  and  draw  the  line  T(7,  which 
represents  the  bottom  of  the  thread.     The  remainder  of  the 
screw  should  now  be  drawn  without  any  trouble. 

26.  Fig.  3  is  a  single  V-tlireaded  screw  1^  inches  in 
diameter  and  having  7  threads  to  the  inch;  that  is,  the 
pitch  is  1  of  an  inch.  Draw  a  cylinder  1]-  inches  in  diam- 
eter, having  ;// //  for  the  center  line.  Lay  off  A  B,  B  C, 
CD,  etc.  each  equal  to  the  pitch,  or  \  inch.  Do  the  same 
on  the  left-hand  side.  By  the  aid  of  the  T  square  and 
60°  triangle  make  the  angles  A  OB,  BO'  G,  etc.  The  rest 
of  the  thread  can  be  drawn  by  referring  to  the  figure. 


18  .  MECHAXICAL  DRAWING  .§  U 

27.  Fig.  4  represents  a  screw  exactly  like  the  preceding 
one,  except  that  the  thread  is  left-handed  instead  of  right- 
handed,  as  in  the  previous  case. 

To  ascertain  whether  a  thread  is  left-  or  right-handed, 
hold  the  screw  in  such  a  position  that  its  axis  is  horizontal. 
If  the  thread  is  right-handed,  as  it  usually  is,  the  angle  that 
the  edge  of  the  thread  makes  with  the  horizontal  on  the 
right-hand  side  is  obtuse ;  if  left-handed,  it  makes  an  acute 
angle  Avith  the  right-hand  side  of  the  horizontal.  Xo  fur- 
ther instruction  should  be  necessary  for  drawing  the  thread. 

28.  Fig.    0    represents    a    double    V-tlirea<led    scre^v 

1:^  inches  in  diameter.  It  has  3^  threads  per  inch;  that  is, 
the  pitch  is  1  inch  -^  3|^  =  f  inch.  The  same  remarks 
regarding  the  drawing  of  it  apply  here  that  were  used  in 
describing  Figs.  2  "nd  3. 

29.  Fig.  6  reprc.-ents  a  section  of  a  brass  nipple. 
When  the  diameter  of  i  nipple  is  given,  the  inside  diameter 
is  always  meant,  unless  otherwise  specially  stated.  The 
actual  diameter  of  a  nipple  or  pipe  is  very  rarely  given,  but 
must  be  taken  from  printed  tables.  The  nominal  diameter 
of  the  nipple  shown  in  the  figure  is  1  inch,  but  the  actual 
inside  diameter  is  1.05  inches;  from  the  table,  the  outside 
diameter  is  found  to  be  1.32  inches,  making  the  thickness 
.135  of  an  inch.  Owing  to  the  thinness  of  the  shell,  pipe 
threads  are  finer  than  the  threads  on  the  same-sized  rods. 
The  coarsest  pipe  thread  is  8  threads  per  inch.  The  number 
of  threads  per  inch  on  the  nipple  shown  is  11^.  The  thread 
is  tapered  to  make  a  tight  fit,  and  the  length  of  the  threaded 
part  on  each  end  is  0.52 -|- 0.53  =  1.05  inches,  of  which 
length  the  distance  between  a  and  b  represents  the  perfect 
thread,  while  from  b  to  c  the  thread  is  chamfered;  that  is,  it 
dies  out  gradually.  To  draw  the  nipple,  make  a  sectional 
view  as  shown.  Draw  a  cylinder  1.32  inches  in  diameter 
and  having  ;//;/  as  a  center  line;  lay  oflf  the  inside  diameter 
equal  to  1.05  inches  and  draw  A  B  and  CD.  Xow  lay  off 
the  diameter  H K  equal  to  1.2G  inches.  Then  lay  off  the 
distances  H f  and  KL  equal  to  0.52  +  0.53=  1.05  inches; 


§  14  MECHANICAL  DRAWING  19 

join  the  points  H  and  /,  and  K  and  Z,  by  straight  lines 
representing  the  top  of  the  threads.  Now,  on  the  center 
Hne  111 ;/,  or  on  any  Hne  parallel  to  it,  lay  off  a  distance  of 
1  inch  from  the  line  A  C  downwards.  Divide  this  distance 
into  \\\  parts  by  means  of  the  method  given  in  Geometrical 
Drawing.  Project  the  points  just  found  upon  ///,  and,  by 
means  of  the  T  square  and  00°  triangle,  draw  the  threads 
from  //  to/  as  though  all  the  threads  were  perfect.  Draw 
the  threads  on  K L  in  the  same  manner,  remembering  that 
the  divisions  on  ///  are  to  be  advanced  half  a  thread,  as 
shown ;  that  is,  the  top  of  one  thread  and  the  bottom  of  the 
preceding  thread  on  the  other  side  will  be  on  a  horizontal 
line.  Now  lay  off  the  distance  ab  equal  to  0.52  inch  and 
project  it  on  lines  drawn  parallel  to  K L  and  HJ  and 
touching  the  bottom  of  the  threads.  From  the  points  of 
intersection  draw  straight  lines  to  /  and  Z,  in  order  to 
obtain  the  bottoms  of  the  imperfect  threads  extending 
from  b  to  c.  Complete  the  rest  of  the  drawing  in  the  same 
manner. 

.30.  Fig.  7  shows  another  method  of  representing  a 
V-threaded  screw.  This  method  has  the  advantage  of  ma- 
king a  neat-looking  drawing  and  of  being  very  rapid  in 
delineation.  The  pitch  is  laid  off  as  in  the  three  preceding 
figures.  The  heavily  shaded  lines  represent  the  bottom  of 
the  thread  and  their  lengths  are  determined  by  construct- 
ing an  equilateral  triangle  on  the  pitch  distance,  as  shown, 
and  limiting  the  line  to  distances  between  two  correspond- 
ing vertexes  of  the  triangle.  The  diameter  of  the  screw  is 
1  inch  and  the  number  of  threads  per  inch  is  8. 

81.  Fig.  8  represents  the  same  screw  shown  in  Fig.  7, 
but  the  lines  indicating  the  bottom  of  the  thread  are  left 
out  altogether.  This  method  is  used  on  drawings  where 
haste  is  necessary.  Unless  in  very  much  of  a  hurry,  the 
method  shown  in  Fig.  7  is  to  be  preferred.  Ordinarily, 
when  drawing  screws  as  represented  by  Figs.  7  and  8,  it  is 
not  customary  to  lay  off  the  pitch  and  the  depth  of  the 
thread  as  above  mentioned — the  distances  between  the  lines 
M.  E.     v.— 8 


20  MECHANICAL  DRAWING  §  14 

representing  the  threads  are  simply  gauged  by  the  eye ;  prac- 
tice will  enable  this  to  be  done  very  quickly  and  accurately. 

32.     Fig.   9   shows    two  views  of  a  small    hand,  wheel. 

To  draw  it,  locate  the  center  O,  and  through  O  draw  the 
center  lines  /  f  and  ///  ;/  at  right  angles  to  each  other.  From 
(7  as  a  center,  with  the  compasses  set  to  the  radius  of  the 
wheel,  or  3^  inches,  draw  the  outer  circle  of  the  rim;  then, 
lay  off  the  thickness  of  the  rim.  which  is  f  of  an  inch,  and 
draw  the  inner  circle.  Through  O,  with  the  T  square  and 
60°  triangle,  draw  the  center  lines  A  B  and  C  D.  With  O 
as  a  center,  draw  two  circles,  one  having  a  diameter  of 
f  inch,  to  represent  the  hole,  and  the  other  a  diameter  of 
1^  inches,  to  represent  the  outside  of  the  hub.  To  draw  the 
arms,  make  the  chords  of  the  arcs  a  b,  c  li,  etc.  each  h  inch 
long.  Make  the  chords  h  I,  i  k,  etc.  each  f  inch  long  and 
draw  //',  //  a,  k  i/,  etc.  With  a  radius  equal  to  ^  of  an  inch, 
describe  the  fillets,  or  arcs.  A'  B\  C  D\  etc.  tangent  to  the 
arms  at  A' ,  B\  C,  etc.  With  a  radius  equal  to  ^  of  an  inch, 
describe  the  fillets  or  arcs  tangent  to  the  inside  of  the  rim 
and  to  the  arms.  All  these  arcs  terminate  at  the  point 
of  tangency.  The  cross-section  part  on  the  arm  indicates 
that  a  cross-section  taken  at  £  F  would  look  as  shown ;  that 
is,  that  the  arm  is  elliptical. 

The  other  view  shows  a  conventional  method  largely 
used  in  drawing  rooms  of  indicating  a  section  of  the  wheel. 
It  is  termed  a  conventional  method  because  it  would  really 
be  impossible  to  obtain  a  section  like  the  one  shown.  Theo- 
retically, the  arm  in  this  view  should  be  sectioned,  but,  for 
convenience,  a  section  is  imagined  to  be  taken  through  the 
rim  and  hub  on  the  line  /  z-  and  turned  around  to  the  posi- 
tion /;/ ;/,  and  the  two  arms  are  shown  in  projection  as  if 
they  were  directly  back  of  the  line  ;//  ;/.  Draw  the  center 
line  /  (/  and  the  sections  of  the  rim  as  shown.  Draw  the 
arms  from  the  dimensions  given.  Draw  the  hub  as  shown, 
using  a  radius  of  3%  of  an  inch  to  draw  the  fillets  or  arcs 
tangent  to  A  B  and  to  the  arm;  make  O'  H'  equal  to  O  H 
in  the  other  view  and  describe  an  arc  through  //'  tangent 


MACHINE 


RT,/:ETZD  JCI77T. 

W-rp-u/yht  I-^c^i    J^uZl  Sz^e 


H  ^i'-'^i&i 


JU7:E25.  /3SS 


For  notice  of  copvriglit,  see  pagij 


DETAILS 


J=-LO.  S 


CLAMPBOX 

Ccx^t  Iron,    J-Ca.Z/Svze 


mediately  followinsr  the  title  pasre 


J-OHN  3MIT?-C.CLAS3  M245e9. 


§  U  MECHANICAL  DRAWING  21 

to  the  two  arcs  just  clrciwn,  which  are  tangent  to  A  B  and 
the  arm.  The  rest  of  the  drawing  can  then  be  completed 
without  further  explanation. 

33.  Fig.  10  shows  a  crank,  which  should  be  drawn 
without  difficulty  from  the  dimensions  given.  The  pin  is 
forced  in  and  the  end  riveted  over  at  A  B,  to  prevent  it 
from  being  pulled  out. 

DKAWING  P]LATE,  TITI.E  :    3IACHI:N"F.  DETAILS 

34.  Fig.    1   shows    a   double   sqxiare-tlireaded   scre\v 

3  inches  in  diameter  and  1^  inches  pitch,  having  a  hexag- 
onal head  and  nut.  The  screw  is  10-J-I  inches  long  and  the 
distance  between  the  faces  of  the  head  and  nut  is  G^f  inches. 
The  head  and  the  nut  are  each  2\^  inches  long,  are  hexago- 
nal in  shape,  and  their  long  diameters  are  5^^  inches.  Since 
this  screw  is  drawn  full  size  and  is  comparatively  of  large 
diameter,  the  true  projection  of  the  helix  is  drawn  instead 
of  the  conventional  method  used  in  Fig.  2  of  the  plate 
entitled  Details.  Each  thread  terminates  in  a  ^''-g-inch  hole 
under  the  shoulder  of  the  head. 

Draw  the  center  line  ;// ;/ ;  lay  off  ^  ^'  equal  to  6J|  inches, 
and  through  A'  draw  CD  perpendicular  to  ;//  n.  Make  A'  £ 
equal  to  2|f  inches  and  draw  F  G  parallel  to  C  D.  Divide 
C  D^  which  is  5:^  inches  long,  into  4  equal  parts,  and  through 
the  points  (T,  i/,  /,  and  D  draw  lines  C  L,  HI,/  K,  and 
D  M  parallel  to  ;//  ;/.  With  A'  as  a  center  and  A'  E  as  a. 
radius,  describe  the  arc  /  E  K,  intersecting  //  /  and  /  K 
in  /  and  K.  Project  the  points  /and  K  upon  C  L  and  D  M\ 
locating  the  points  L  and  M,  and  pass  through  L  and  /  and 
through  K  and  M  circular  arcs  tangent  to  F  G.  The 
centers  of  these  arcs  will  lie  on  lines  parallel  to  the  center 
line  in  n  and  midway  between  L  and  /  and  between  K 
and  M.  Therefore,  draw  N  G  parallel  to  the  center  line  ;//  n 
through  a  point  half  the  distance  between  /  K  and  D  J/, 
and  find  by  trial  a  point  A^  at  such  a  distance  from  G  that, 
with  the  radius  N  G,  the  arc  will  pass  through  /v",  G,  and  J/. 
Draw  the  nut  in  the  same  manner.  This  is  the  conven- 
tii)nal  method  of  representing  a  nut. 


22  MECHANICAL  DRAWING  §  14 

To  construct  the  screw,  draw  through  some  convenient 
point  O  a  line  Q  R  parallel  to  C  D.  From  O  as  a  center, 
describe  two  semicircles  whose  diameters  are  equal  to  the 
top  and  bottom  diameters  of  the  screw  thread,  intersecting 
the  line  Q  R  va.  the  points  O^  q,  r,  and  R.  From  these 
points  draw  lines  to  S,  s,  T,  and  /  parallel  to  the  center 
line  sn  n ;  e\ndently  these  lines  coincide  with  the  top  and 
f  the  screw  thread.  The  depth  of  the  thread  is 
:  ~  in  Fig.  2  of  the  plate  entitled  Details. 

Divide  the  semicircles  into  any  number  of  equal  parts,  in 
this  case  6 ;  this  is  best  done  by  first  dividing  the  exterior 
semicircle  into  the  required  number  of  parts,  as  (7,  i,  2,  3, 
4,  5,  and  R,  and  then  drawing  radii  through  each  point 
from  O,  which  divides  the  interior  semicircle  similarh.  Lay 
oflE  on  the  line  R  t,  or  any  other  line,  as  U  J ',  which  is  par- 
allel to  the  center  line  ///  n,  the  pitch  of  the  screw,  or 
1^  inches,  and  divide  it  into  twice  the  niunber  of  equal  parts 
that  the  semicircles  have  been  divided  into.  Locate  the 
point  Fat  a  distance  of  ff  inch  from  the  shoulder  C D  oi 
the  head. 

Now,  construct  the  helix  6S-Jf-S-2—l-W iox  the  top  diam- 
eter; also  the  helix  6'S—^^-X\  having  the  same  pitch,  for 
the  bottom  diameter.  As  this  is  a  double  square-threaded 
screw,  the  widths,  as  well  as  the  depths,  of  the  threads  and 
grrooves  are  equal  to  \  of  the  pitch,  or  1  i  X  i  =  f  of  an  inch. 
(See  Fig.  2,  last  plate.) 

Consequently,  to  lay  off  the  widths  of  the  threads  and 
grooves,  divide  the  pitch,  or  line  U  F,  into  4  equal  parts,  as 
1,  2^  3,  and  6',  and  through  these  points  draw  lines  parallel 
to  CD,  intersecting  the  lines  that  represent  the  top  and 
bottom  diameters  of  the  screw  in  the  points  ^^,  e,  and  Z'  \ 
b^  g,  and  Z\  r,  f,  and  W\  d,  X,  and  F;  draw  through  these 
points  other  helical  curves  parallel  to  those  alread\-  drawn. 
The  remaining  threads  of  the  screw  ma}-  be  drawn  in  the 
same  manner,  but  an  easier  and  quicker  method  is  to  cut  a 
curve,  of  the  same  shape  as  those  already-  drawn,  out  of  a  piece 
of  bristol  board  or  cardboard  and  use  it  as  a  template.  The 
manner  of  constructing  the  curves  X-l'-2'-S'  and  3-^'-o'-G' 


§  U  MECHANICAL  DRAWING  23 

should  be  clearly  evident  from  the  drawing.  The  dotted 
curves  need  not  be  drawn  by  the  student,  as  they  are 
merely  put  in  to  show  the  connections  between  the  different 
threads. 

t>«5.  In  Fig.  2  are  shown  a  section  and  end  view  of  a 
shaft  flangre  coiiplinj?,  used  for  connecting  the  ends  of 
two  shafts.  In  order  not  to  take  up  too  much  room  on  the 
plate,  the  coupling  is  drawn  to  a  reduced  scale,  which  is 
marked  on  the  drawing  as  3"  =  1  ft.  Using  the  scale  of 
3"  =  1  ft.  (see  Art.  21),  draw  the  figure  from  the  dimensions 
given,  first  drawing  all  of  the  sectional  part,  except  the  bolt. 
Then  draw  the  end  elevation  as  shown,  and,  lastly,  the  bolt 
in  the  sectional  view.  The  part  A  B  C  is  a  key  sunk  into 
the  shafts  to  keep  them  from  turning  within  the  coupling. 
The  shafts  themselves  are  made  of  wrought  iron,  as  indicated 
by  the  sectioning. 

36.  Fig.  3  is  a  front  and  side  elevation  of  a  gland. 
This  is  also  drawn  to  a  scale  of  3"  =  1  ft.,  and  should  be 
constructed  without  difficulty  from  the  dimensions  given. 
The  line  OA,  marked  Of "  r,  means  that  the  radius  of  the 
arc  BAD  is  6|  inches  long;  so,  also,  CB,  marked  11"  r, 
means  that  the  radius  of  the  arc  B£  is  1^  inches  long. 
Since  no  dimensions  are  given  for  the  other  two  arcs,  it  is 
understood  that  their  radii  are  the  same  as  for  the  first  two. 
Whenever  a  dimension  is  given,  like  0^1,  specifying  the 
length  of  a  radius,  the  letter  r,  or  the  abbreviation  rad., 
should  always  be  placed  after  it. 

37.  Fig.  4  shows  a  riveted  joint,  or  two  plates  riveted 
together.  The  front  elevation  and  a  cross-section  through 
one  of  the  rivets  are  given.  The  diameter  of  the  rivets  is 
^  inch  and  the  pitch  (the  distance  between  the  centers  of 
two  consecutive  rivets)  is  14-  inches.  These  and  the  other 
dimensions  are  obtained  from  the  drawing.  Draw  the  cross- 
section  first,  as  shown  in  the  figure,  and  after  that  the 
elevation.     The  scale  is  full  size. 


U  MECHANICAL  DRAWING  §  U 

38.  Fig.  5  shows  a  clanii},  clog,  or  carrier,  as  it  might 
be  termed.  It  can  be  readily  drawn  from  the  dimensions 
given.  It  will  be  noticed  that  part  of  the  pin  A  B  is  flat- 
tened. This  is  seen  more  clearly  in  the  top  view,  shown  by 
the  dotted  lines.      It  is  to  be  drawn  to  a  scale  of  3'  =  1  ft. 

39.  Fig.  6  shows  a  clamp  box  to  be  attached  to  beams 
for  shafting  to  pass  through.  The  scale  to  which  it  is  drawn 
is  half  size,  that  is,  6"  =  1  ft.  The  curves  at  A,  B,  etc.  are 
not  exact  projections  of  the  curve  of  intersection  of  the 
round  and  flat  surfaces  shown  in  the  other  view,  but  are 
drawn  as  circular  arcs,  from  which  they  differ  but  little,  and 
the  time  occupied  in  finding  the  different  points  on  the  curve 
is  saved.  This  answers  the  purpose  when  making  shop  draw- 
ings just  as  well  as  the  exact  method.  The  curve  C F D  is 
the  projection  of  the  part  D  E  F,  shown  by  the  dotted  lines 
in  the  elevation,  which  is  removed  in  order  that  the  nut 
may  be  turned  round.  G  is  an  oil  hole.  It  will  be  noticed 
that  the  bolt  holes  are  larger  than  the  bolts;  this  allows  the 
clamp  box  a  little  play,  should  it  be "  necessary.  It  also 
allows  the  holes  to  be  cored  in  the  box  when  the  casting  is 
made,  instead  of  being  drilled  afterwards.  On  all  drawings 
made  for  the  shop  or  on  which  dimensions  are  given  or 
required,  the  scale  should  invariabh'  be  specified.  If  more 
than  one  scale  is  used,  as  in  the  last  plate,  where  three  dif- 
ferent ones  were  used,  it  should  be  given  for  each  figure. 


DRAWIXG   PliATE,  TITLE  :     FI^AJSTGE   COI'PEIXG 

JrO.  This  plate  shows  a  drawing  of  a  flange  coupling 
suitable  for  connecting  two  lengths  of  2-inch  line  shafting. 
Fig.  1  is  a  section  on  the  line  A  B  (Fig.  2)  and  shows  how 
the  two  parts  C  and  D  of  the  coupling  are  bolted  together 
through  their  flanges,  hence  the  name  flange  coupling. 
Each  part  is  keyed  on  its  shaft  separately  and  true  alinement 
of  the  shafts  is  insured  by  means  of  the  recess  in  C  into  which 
is  fitted  a  raised  boss  on  D.  The  two  parts  of  the  coupling 
are  first  bored  and  then  faced  up  on  the  surfaces  EFGHIJ. 


FLANGE  I 


-_7/ 


JLOVF  2c,  J593. 


For  notice  of  copyright,  see  page  il 


DUPLINB 


'ZSize. 


Fvq  e 


ediately  following  the  title  page. 


JOM//  SMITH,  CLASS  H^  4529. 


§  U  MECHANICAL  DRAWING  25 

They  are   then   clamped  together  and  the  keyway  is  cut. 
Fig.  2  is  a  half-end  view  and  needs  no  comment. 

41.  To  begin  the  drawing,  draw  the  horizontal  center 
line  xj'  G|  inches  from  the  lower  border  line.  Draw  Fig.  1 
first,  commencing  with  the  vertical  joint  line  E/  of  inches 
from  the  left-hand  border  line.  It  is  now  well  to  draw  the 
shaft.  This  is  2  inches  in  diameter;  therefore,  lay  off 
vertically  on  each  side  of  .vy,  2  -=-  2  =  1  inch,  and  through 
the  points  thus  located  draw  the  horizontal  lines  ^^?' and /;//. 
Next  draw  the  hub  of  the  coupling.  Lay  off  3^  inches  hori- 
zontally on  each  side  of  the  vertical  joint  line  £/  and 
draw  cc'  and  dd'.  The  diameter  of  the  hub  is  4|-  inches; 
therefore,  lay  off  4^  -=-  2  =  2\  inches  on  each  side  of  xj'  and 
draw  t'e'  and /"_/';  then  draw  in  the  round  corners  c'  e,  c'  d\ 
df\  and  r/"  with  a  radius  of  \  inch.  To  the  left  of  the  joint 
line  EJ  lay  oft"  a  distance  of  ^  inch  and  draw  H  G\  lay  oft" 
3-4-2=1^  inches  on  each  side  of  x y  and  draw  6^/^  and  HI. 
On  referring  to  Fig.  2,  it  will  be  found  that  the  outside 
diameter  of  the  coupling  is  9 J  inches;  lay  off  half  this  diam- 
eter, or  4:|-  inches,  on  each  side  of  the  horizontal  center 
line  X y  and  draw  g g'  and  Ji  k'.  Each  flange  is  If  inches  in 
width  on  the  outer  face;  lay  off  If  inches  to  both  right  and 
left  of  E J  and  draw  the  vertical  lines  ii'  and 77',  putting  in 
the  rounded  corners  i,  i' ,  etc.  with  the  compasses  set  to 
^  inch,  in  each  case  completing  the  circle  faintly.  Now, 
draw  faint  vertical  lines  k k'  and  //'  each  -if  inch  from  the 
vertical,  joint  line  EJ.  Make  k  k'  and  //'  each  equal  to 
8|  inches;  to  do  this  lay  off  4:^  inches  vertically  on  each  side 
of  xy  and  draw  faint  construction  lines  /''  /',  k  I.  Then  with 
the  compasses  set  to  a  radius  of  /^  inch  and  a  center  on  the 
line  /'/''  produced,  draw  the  semicircle  k'  mil.  Draw  no 
just  touching  this  semicircle  and  tangent  to  the  dotted  circle 
mentioned  above;  then  draw  the  other  three  similar  parts 
of  the  flange  in  the  same  manner.  Now,  by  reference  to 
Fig.  2,  locate  the  bolt  center  lines  //'  and  qq' ;  draw  the 
bolts  and  nuts  and  complete  Fig.  1  from  the  dimensions 
given. 


26  MECHANICAL  DRAWING  §  14 

The  reference  letters  printed  in  bold-face  italics  should  be 
omitted  on  the  drawing  made  by  the  student. 


I>RA^T::N^G  PI^^TE,   title:    ECCEXTRIC  AZS'D 
BRAKE  LEA'ER 

42.     Fig.    1    shows   an    elevation   of    an    eccentric    and 

its  strap.  The  strap  is  made  in  two  pieces  and  bolted 
together  with  a  small  space  gV  inch  wide  between  them. 
Locate  the  point  O,  the  center  of  the  strap,  and  draw  the 
center  lines  A  B,  m ;/,  C  C\  and  D  D' .  Make  the  offset  O  0' 
one-half  of  the  throw  of  the  eccentric,  or,  in  this  case,  |-  inch, 
thus  locating  0\  the  center  of  the  eccentric  shaft.  Con- 
struct the  rest  of  the  view  from  the  dimensions  given,  noting 
that  the  arcs  E  E'  and  EP  are  concentric  with  O,  while  G  G' 
and  H H'  are  concentric  with  O'.  The  part  F F  G'  G  is 
entirely  open  and  is  made  so  in  order  to  lighten  the  eccentric. 

•43.     Fig.   2   is    a    section  of   the  eccentric   an<l   strap. 

The  section  is  drawn  in  a  conventional  manner,  it  being  all 
taken  on  the  line  A  B,  Fig.  1,  except  that  part  of  the  eccen- 
tric between  G'  and  F,  where,  instead  of  sectioning,  the  draw- 
ing shows  it  open  from  L  to  P,  Fig.  2.  This  attracts  attention 
to  the  open  part  of  the  eccentric  and  sliows  it  more  clearly. 
It  will  be  noticed  that  the  slope  of  the  threads  in  the  sec- 
tional part  is  the  same  as  for  a  left-handed  screw.  The 
thread,  however,  is  right-handed,  and  the  reason  for  show- 
ing it  in  this  manner  is  that  it  is  the  bottom  of  the  thread 
that  is  being  looked  at;  that  is,  the  section  of  a  right- 
handed  nut  is  the  same  as  the  projection  of  the  top  of  a 
left-handed  screw.  It  will  also  be  noticed  that  the  sectional 
lines  on  the  eccentric  run  in  opposite  directions  to  those  on 
the  strap.  This  is  always  done  when  two  different  pieces 
meet,  and  serves  to  indicate  that  they  are  separate  pieces. 
Each  piece  should  be  sectioned  entirely  in  the  same  direction, 
no  matter  if  there  is  a  break,  as  in  the  present  case,  between 
L  and  P.  This  shows  that  A  B  C  D  E  is  one  piece.  The 
dotted  hole  at  A',  Fig.  1,  is  an  oil  hole. 


tH 


ECCENTRIC  AN1 


.^^ 


Ol23C^  TOTi^. 


Fi^J 


m.. 


..!*_ 


^-^-^- 


z      ^^'^ 


rt-p  2 


JujiE  es./ssa 


:<iice  of  c»pyriglit,  see  page  in 


BRAKE  LEVER. 


H 


H5 


1 


"^""^^ 


1^- 


ffl 


BRAJKE  LEVER 

l/Vrozj.oHt  Irorv. 
Sca^le6=/ft. 


-^.. 


Fi^y 


iiately  following  the  title  page. 


jo^M  syyriTJ^,  CLASS  y/2  45a9 


§  U  MECHANICAL  DRAWING  27 

•44.     Fig.   3  shows  a  brake  lever  drawn  to  a  scale  of 

3  inches  =  1  foot.  Owing  to  its  length  being  too  great  to  be 
shown  entirely  on  the  drawing  to  this  scale,  the  handle  is 
shown  as  if  a  piece  had  been  broken  out,  the  dimension  line, 

4  feet  7  inches,  giving  the  distance  between  the  two  centers  O 
and  O'.  The  lever  should  be  readily  drawn  from  the  dimen- 
sions given.  To  proportion  it  properly,  where  the  size  of 
the  paper  does  not  permit  the  whole  of  it  to  be  drawn,  pro- 
ceed as  follows:  The  length  between  the  centers  is  4  feet 
7  inches  =  55  inches.  The  width  through  O'  is  '2\  inches 
and  through  O  4  inches.  Hence,  4"  -  2 J"  =  If";  1.75" 
=  the  taper  in  55  inches.     Measure  o^  O  A  =  say,  2  feet. 

1  75 
The  width  at  A  may  be  found  as  follows:  -^  =  taper  in 

L  inch.      -^-^  X  24  =  .76  inch,  nearly,   the  taper  in  2  feet. 
5o 

4" —  .76"  =  3.24"  =  ^  r,   or   the    width    at  A.      Now  locate 

the   point    O'  and   from   it   as   a   center   describe   a  curve 

•l^  inches  in  diameter.      Draw  lines  tangent  to  this  curve 

and  parallel  to  the  edges  between  A  and  O  already  found. 

It  should  be  noticed  that  the  center  of  the  brake  lever  in 

the  left-hand  view  is  not  situated  at  the  joint  where  the  two 

parts  come   together,  but  coincides  with  the   center  of  the 

handle,  as  it  should  do. 


DRAWIXG    PLATE,  TITJLE :    TIMBER    TRESTLE 

45.  This  plate  shows  a  drawing  of  a  standard  framed 
timber  trestle  as  used  on  .the  Cleveland  and  Canton 
Railroad.  It  is  called  a  eompound  trestle,  its  various 
members  being  built  up  of  comparatively  small  timbers. 
The  advantages  of  this  style  of  construction  are  the 
reduction  of  the  cost  of  timber  and  the  facility  and  safety 
with  which  repairs  can  be  made.  As  the  parts  are  bolted 
together,  a  defective  piece  may  be  readily  replaced ;  and  the 
pieces  being  separated  from  each  other,  there  is  a  thorough 
circulation  of  air  throughout  the  structure,  which  seasons 
and  preserves  the   timber.      It   is  desirable  in  a  structure  of 


28  MECHANICAL  DRAWING  §  U 

this  kind  to  show  three  views — end  view,  side  view,  and  top 
view  or  plan.  It  is  found  that  the  limits  of  the  drawing 
will  admit  of  the  figures  being  drawn  to  a  scale  of  f  inch 
=  1  foot ;  therefore,  this  scale  will  be  adopted. 

The  drawing  should  be  made  from  the  written  dimensions 
given  on  the  plate;  and  when  drawing  Fig.  1,  reference 
must  frequently  be  made  to  Fig.  2  for  some  of  the  dimen- 
sions required. 

46.  To  begin  the  drawing,  draw  the  vertical  center  line 
a  a  at  a  distance  of  4^  inches  from  the  right-hand  border 
line.  Draw  the  horizontal  base  line  b  b'  2X  a  distance  of 
1^  inches  above  the  lower  border  line.  From  and  above  the 
line  b  b'  lay  off  a  distance  of  I'l  inches  on  the  vertical  center 
line  a  a' ,  and  through  the  point  thus  laid  off  draw  the  hori- 
zontal line  c  c' .  On  each  side  of  the  center  line  a  a'  lay  off 
on  the  horizontal  base  line  a  distance  of  9  feet  6  inches  and 
draw  the  short  vertical  lines  b  c  and  b'  c'  representing  the 
ends  of  the  ground  sill.  From  and  above  the  line  c  c'  lay 
off  on  the  vertical  center  line  a  a  a  distance  of  VI  feet, 
which  is  the  distance  from  the  upper  edge  of  the  ground  sill 
to  the  lower  edge  of  the  cap,  and  through  the  point  thus  laid 
off  draw  the  horizontal  line  d d'  \  at  a  distance  of  1'2  inches 
above  this  line  draw  the  line  c  c  representing  the  upper  edge 
of  the  cap.  On  each  side  of  the  center  line  a  a'  lay  off  on 
the  horizontal  line  d d'  one-half  the  length  of  the  cap,  or 
12  -=-2  =  6  feet,  and  draw  the  vertical  lines  d c  and  d'  e  reprje- 
senting  the  ends  of  the  cap.  Next  draw  the  upright  or 
plumb  posts  between  the  ground  sill  and  the  cap;  on  each 
side  of  the  center  line  a  a'  lay  off  on  the  horizontal  line  /;  b' 
a  distance  of  2  feet  6  inches,  and  through  the  points  thus 
laid  off  draw  the  vertical  center  lines  ff  and  g g  of  the 
plumb  posts.  On  each  side  of  the  center  line  f  f  lay  off  on 
the  horizontal  line  /;  /; '  one-half  the  width  of  the  post,  or 
12  -=-2  =  6  inches,  and  draw  the  vertical  lines  //  h'  and  jj'. 
Draw  the  other  plumb  post  in  a  similar  manner.  From  the 
point  c  on  the  ground  sill  and  towards  the  center  line  a  a  lay 
off  on  the  line  c  c'  a  distance  c  k  of  18  inches;  next,  from  the 


5== 

-   -   r 

-  r 

F  r  ^ 

: T — 

^ 

c;                              3 

3 

3      r^ 

9~9 
©"9 

a      g    1 

c 

tf 

Julin.  2o,  Jo  93. 


TIMBER  TRESTLE 


lediately  following  the  title  page. 


JOJi:N  SMITJ{,  CLAS3  N^4529. 


§  14  MECHANICAL  DRAWING  29 

point  e  on  the  cap  and  towards  the  center  line  a  a'  lay  off  a  dis- 
tance e  k'  of  lo'inches;  join  the  points  /'  and  /-'.      Now,  with 
the  aid  of  a  triangle  draw  a  line  in  lead  pencil  intersecting 
and  at  right  angles  to  the  line  /'/■',    and  on  this  line   lay 
off  towards   the    center  line  a  a'  and    from    the  line  k  k'  a 
distance    equal    to   the    width    of   the    post,    or    12    inches; 
through  the  point  thus  laid    off  draw  the   line  //'  parallel 
to  k  k'.    This  will  complete  the  left-hand  batter  post.    Draw 
the  right-hand  batter  post  in   a  similar  manner.     We  are 
now  ready  to  draw  the  3"  X  10"  sway-braces  which  connect 
the  opposite  ends  of  the  ground  sill  and  cap.      From  and  to 
the  left  of  the  lower  right-hand  corner  b'  of  the  ground  sill 
layoff  a  distance  b'  in  of  5  inches;  ixoxw  and  below  the  upper 
left-hand    corner    c   of    the   cap    lay  off    a  distance  e  in'   of 
G  inches;  join  the  points  ni  and  vi  \  then  draw  a  line  parallel 
to  VI  ni  at  a  distance  from  it  equal  to  the  width  of  the  sway- 
brace— that  is,   10  inches.      Draw  the  brace  connecting  the 
other  ends  of  the  sill  and  cap  in  a  similar  manner.     The 
frame  so  far  drawn  is  called  the  bent.     There  is  a  space  of 
15  feet  from  the  center  of  one  bent  to  the  center  of  the  next, 
as  shown  in  Fig.  2,  and  they  are  connected  from  cap  to  cap 
by  stringers.     In  Fig.  1  these  stringers  are  shown  in  sec- 
tion, three  being  set  side  by  side  on  the  cap  over  each  plumb 
post.      Each  stringer  is  notched  1  inch  over  the  cap.     From 
and   below  the  line  c  c'  lay  oft"  a  distance  of  1  inch  and  draw 
a  horizontal  line  representing  the  position  of  the  lower  edges 
of  the  stringers;  from  and  above  this  line  lay  off  a  distance 
equal  to  the  depth  of  the  stringers,  or  14  inches,  and  draw 
another  horizontal  line  representing  the  upper  edges  of  the 
stringers.      On  each  side  of  the  plumb  post  center  line  //' 
lay  off  on  the  line  c  c'  one-half  the  thickness  of  a  stringer, 
or  7  -^-  2  =  ;3.V  inches;  then,  through  the  points  thus  laid  off 
draw  the  vertical  lines;/;/'  and  oo\  completing  the  middle 
stringer.      To  the  left  and  right  of  this  stringer,  at  a  dis- 
tance of  2i^  inches  from  the  faces  ;/;/'  and  oo,  respectively, 
draw  vertical  lines  representing  the  inner  faces  of  the  other 
two  stringers.      Lay   off    7    inches    to    the    left   and    right, 
respectively,  of  these  lines,  and  through  the  points  thus  laid 


30  MECHANICAL  DRAWING  g  14 

off  draw  vertical  lines  completing  the  two  outer  stringers. 
In  a  similar  manner  draw  the  stringers  over  the  right-hand 
plumb  post.  The  stringers  are  kept  in  position  sidewa}^s  by 
three  brace-blocks  bolted  to  the  cap ;  to  draw  these  lay  off 
from  and  above  the  top  edge  e  e'  of  the  cap  a  distance  of 
3  inches  and  draw  the  horizontal  line  pp' .  The  center 
brace-block  fits  between  the  two  inner  stringers,  as  shown  ; 
the  outer  ones  are  20  inches  long;  lay  off  this  length  to  the 
right  and  left  of  the  right-  and  left-hand  outer  stringers 
and  draw  the  short  vertical  lines  completing  the  brace- 
blocks.  Across  the  stringers  are  laid  the  ties,  which  are 
notched  1  inch  over  the  stringers.  From  and  below  the  top 
edge  11'  o'  of  one  of  the  stringers  lay  off  a  distance  of  1  inch 
and  draw  the  horizontal  line  zz'  representing  the  lower  edge 
of  the  tie  notched  over  the  stringers,  as  shown ;  from  and 
above  this  line  lay  off  the  depth  of  the  tie,  or  7  inches,  and 
draw  the  line  representing  the  top  edge  of  the  tie.  On  each 
side  of  the  center  line  a  a'  lay  off  on  the  line  z  z'  one-half 
the  length  of  the  tie,  or  8'  4"  h-  2  =  4  feet  2  inches,  and 
draw  the  vertical  lines  representing  the  ends  of  the  tie,  con- 
tinuing these  lines  upwards  for  a  short  distance.  Now,  con- 
necting the  ends  of  the  ties,  in  a  direction  parallel  to  the 
rails,  are  wooden  guard  rails  to  prevent  the  cars  from 
jumping  the  trestle,  if,  by  any  chance,  they  should  leave 
the  tracks.  These  guard  rails  are  notched  1  inch  over  the 
ties.  From  and  below  the  top  edge  of  the  tie  lay  off  a  dis- 
tance of  1  inch  and  draw  a  short  horizontal  line  at  each  end 
of  the  tie;  from  and  above  these  lines  lay  off  a  distance  of 
8  inches  and  draw  horizontal  lines  representing  the  top 
edges  of  the  guard  rails.  From  each  end  of  the  tie  lay  off 
towards  the  center  line  a  a'  2^  distance  of  8  inches  and  draw 
the  short  vertical  lines  to  complete  the  guard  rails,  as 
shown.  The  distance  between  the  inner  flanges  of  the  steel 
track  rails  is  4  feet  8i  inches;  on  each  side  of  the  center 
line  a  a'  lay  off  on  the  top  edge  of  the  tie  one-half  this  dis- 
tance, or  4'  8|"  ^2  =  2  feet  4^  inches,  and  draw  vertical 
lines  in  lead  pencil.  The  rails  may  now  be  drawn  in  and 
should  be  5  inches  high,  5  inches  wide   on  the  lower  flange. 


§  14  MECHANICAL  DRAWING  31 

and  2  inches  wide  on  the  top  flange.  By  referring  to  Fig.  2, 
it  will  be  seen  that  the  bents  are  held  together  at  their  bases 
by  longitudinal  ties;  these  are  called  girts  and  extend  from 
post  to  post,  being  attached  to  the  lower  ends  of  both  batter 
and  plumb  posts.  All  the  girts  are  of  10"  X  4"  section. 
From  and  to  the  right  of  the  line  //'  lay  off  a  distance  of 
4  inches  and  draw  the  line  qq'  parallel  to  the  line  //'.  From 
the  point  of  intersection  of  the  line  qq'  and  the  line  c c' 
draw  a  line  ^r  at  right  angles  to  the  line  //'  of  the  batter 
post.  From  and  above  the  point  r  lay  off  along  the  line  // 
a  distance  of  10  inches  and  draw  q'  r'  parallel  to  the  line  q  ?'. 
In  a  similar  manner  draw  the  girt  at  the  base  of  the  right- 
hand  batter  post  and  then  draw  two  girts  at  the  base  of 
each  plumb  post.  The  bents  are  still  further  secured  by 
means  of  diagonal  braces  which  connect  the  top  of  a  plumb 
post  in  one  bent  to  the  bottom  of  a  plumb  post  in  the  next 
bent,  as  shown  in  Fig.  2;  to  each  6"  X  8"  central  brace, 
which  is  bolted  to  the  post  by  means  of  the  bolts  7',  i'\  there 
are  two  3"  X  8"  braces,  one  on  each  side  of  the  central  brace, 
secured  to  the  posts  by  bolts  tc'  and  to  the  central  brace  by  a 
bolt  zi>".  We  will  first  draw  the  two  3"  X  8"  braces.  From 
each  side  of  the  left-hand  plumb  post  lay  off  a  distance  of 
3  inches  and  draw  vertical  lines  extending  from  the  upper 
edge  of  the  girts  to  the  lower  edge  of  the  cap ;  on  either  of 
these  lines  lay  off  from  and  below  the  line  d d'  a  distance 
of  3  inches  and  draw  the  short  horizontal  lines  representing 
the  ends  of  the  diagonal  braces  seen  in  this  view.  Now 
draw  the  6"  X  8"  brace  by  laying  oif  on  each  side  of  the 
center  line  ff  one-half  the  thickness  of  the  brace,  or 
6-^-2  =  3  inches,  and  drawing  vertical  lines.  This  brace 
extends  1\  inches  below  the  top  of  the  sill  and  li  inches 
above  the  lower  edge  d d'  of  the  cap;  lay  off  these  distances 
and  draw  the  short  horizontal  lines  representing  the  ends  of 
the  brace.  Each  bent  is  supported  upon  four  piles,  two  of 
which  are  located  immediately  under  the  plumb  posts,  the 
centers  of  the  other  two  being  18  inches  from  the  ends  of 
the  sill.  The  tops  of  the  piles  are  tenoned  and  enter  for  a 
depth  of  G  inches  between  the  two  timbers  forming  the  sill. 


32  MECHAXICAL. DRAWING  ^4 

The  student  should  be  able  to  draw  these  piles  without 
special  instructions.  Next  locate  the  bolt  centers;  begin 
by  drawing  the  center  lines  of  the  batter  posts  and  sway- 
braces  in  lead  pencil,  and  from  the  drawing  locate  all  bolts 
on  these  lines.  Where  the  timbers  cross  each  other  and  are 
bolted  together,  draw  in  lead  pencil  the  short  diagonal  of 
the  diamond  formed  by  the  crossing  of  the  two  timbers;  the 
point  where  these  diagonals  meet  is  the  center  of  the  bolt. 
Now  locate  all  bolts  on  the  center  lines /y  and  gg'  and 
then  the  bolts  on  the  center  lines  of  the  two  outside  piles. 
Dimensions  are  given  which  will  enable  the  student  to  locate 
all  other  bolts.  All  the  bolts  are  f  inch  diameter,  with  heads 
and  nuts  f  inch  thick  and  1^  inches  square.  The  washers  are 
all  3  inches  in  diameter,  with  the  exception  of  those  marked 
3|  inches  in  Fig.  2,  and  are  all  f  inch  thick.  In  manj-  places 
washers  are  introduced  between  timbers  to  act  as  separa- 
tors; these  are  all  3  inches  in  diameter,  with  the  exception 
of  those  between  the  stringers,  which  are  o\  inches  in 
diameter. 

4T.  We  are  now  ready  to  proceed  with  Fig.  2.  Draw 
the  vertical  center  line  s s'  2iX.  a.  distance  of  2^  inches  from 
the  left-hand  border  line.  From  and  to  the  right  of  s  s'  lay 
off  a  distance  of  15  feet  and  draw  the  center  line  ft'.  Pro- 
jecting from  the  lower  edge  of  the  sill  in  Fig.  1  draw  short 
horizontal  lines  a  b  and  c  d  and  also  draw  the  lines  cfz.n6.gh 
by  projecting  from  the  top  edge  of  the  cap.  On  each  side 
of  the  center  line  t  f  lay  off  one-half  the  distance  between 
the  two  timbers  that  form  the  sill,  or  3"  ^  2  =  \\  inches, 
and  draw  vertical  lines  between  cd  and  g/i.  To  the  right 
and  left  of  eg  and  dh,  respectively,  lay  off  a  distance  of 
G  inches  and  draw  vertical  lines  as  before.  Projecting  from 
the  top  edge  of  the  brace-block  in  Fig.  1  draw  the  horizontal 
line  J k  and  complete  the  end  view  of  the  brace-block  by 
drawing  the  short  vertical  lines  gj  and  //  /'.  The  ends  of 
the  pieces  forming  the  posts  are  separated  by  tenon  blocks 
3  inches  thick  and  2  feet  8  inches  long.  From  and  below 
the  line  sr h  lav  off  a  distance  of  2  feet   8  inches  and  draw 


§  U  MECHANICAL  DRAWING  33 

the  line  ////.  From  and  above  the  line  r^/lay  off  a  distance 
of  6  inches  and  draw  a  dotted  horizontal  line ;  above  this 
line  lay  off  a  distance  of  2  feet  8  inches  and  draw  the 
line  )io.  In  a  similar  manner  draw  the  sill,  post,  and  cap, 
etc.  about  the  center  line  s s' .  Projecting  from  Fig.  1 
draw  the  horizontal  lines /^  and  r  s"  representing  the  upper 
and  lower  edges  of  the  stringers.  These  stringers  are 
30  feet  long  and  their  joints  are  staggered ;  the  joint  on  the 
center  line  1 1'  is  represented  by  a  full  line,  while  that  on  the 
center  line  j-/  is  a  dotted  line.  Projecting  from  Fig.  1  draw 
in  lead  pencil  horizontal  lines  representing  the  upper  and 
lower  edges  of  the  ends  of  the  ties.  From  and  to  the  right 
of  the  center  line  s  s'  on  the  line  r s"  lay  off  a  distance  of 
3  inches  and  draw  a  vertical  line,  as  shown ;  on  each  side  of 
this  vertical  line  and  along  the  line  r  s"  space  off  distances 
of  15  inches  and  draw  vertical  lines  through  each  of  the 
points;  to  the  right  of  each  of  these  vertical  lines  lay  off 
a  distance  of  9  inches  and  draw  vertical  lines  to  complete 
the  end  views  of  the  ties,  as  shown.  Projecting  from 
Fig.  1  draw  the  upper  edges  vzu  and  lower  edges  t"  u 
of  the  guard  rails,  which  are  notched  1  inch  over  the 
ties,  and  next  draw  the  horizontal  lines  xy,  etc.  repre- 
senting the  longitudinal  girts,  and  then  draw  the  diagonal 
braces,  as  shown.  Now,  to  the  left  of  the  vertical  line  eg 
and  to  the. right  of  dh  lay  off  a  distance  of  3  inches  and 
draw  vertical  lines  defining  the  sway-braces.  Then  draw 
in  the  bolts  from  dimensions  given  and  by  projecting 
from  Fig.  1. 

48.  Next  draw  Fig.  3.  Draw  the  horizontal  center 
line  X  x'  at  a  distance  of  3^  inches  below  the  top 
border  line.  The  student  should  experience  no  diffi- 
culty in  drawing  this  figure;  vertical  lines  should  be 
projected  from  corresponding  lines  in  Fig.  2,  and  all 
measurements  may  be  found  by  reference  to  Figs.  1 
and    2. 

The  reference  letters  printed  in  bold-face  italics  should  be 
omitted  on  the  drawing  made  by  the  student. 


34 


MECHANICAL  DRAWING 


11 


deawtxg  pi^te,  title  :  steel  coeuimxs  ais^d 

cox:nt:ctioxs 

49.  In  this  plate  are  shown  two  designs  of  columns  used 
in  the  construction  of  modern  office  buildings,  and  a  common 
method  of  connecting  the  floorbeams  to  the  columns  is 
shown.  A  side  view  and  a  front  view  of  each  design  is  given, 
and  as  it  would  manifestly  be  impossible  to  draw  the  columns 
in  their  full  length,  and  since,  besides,  it  would  not  serve 
any  useful  purpose  to  do  so,  a  part  of  the  column  between 
the  base  and  the  floorbeam  connection  is  broken  away. 
When  this  is  done,  it  is  understood  that  the  part  broken 
away  is  similar  to  the  ends  of  the  column  next  to  the  break, 
which  in  this  case  is  indicated  by  drawing  a  line  consisting 
of  a  long  dash  and  two  dots  across  the  column.  Among 
mechanical  draftsmen  it  is  customary  to  indicate  a  break  by 
a  wavy  freehand  line,  but  many  architectural  draftsmen 
indicate  a  break  in  the  manner  shown  in  this  plate. 

50.  In  the  design  shown  by  Fig.  1,  the  column  consists 
of  two  channels  placed  back  to  back  and  tied  together  by 
,_^_^  — --,  cover-plates.  A  cross-sec- 
tion of  this  column  is 
shown  in  Fig.  13  (a).  In 
the  design  shown  by  Fig.  2, 
the  column  is  built  up  of 
four  Z  bars  riveted  to  a 
web-plate,  the  column  hav- 
ing the  cross-section  shown 
in  Fig.  13  (d).    The  columns 

in  both  designs  are  made  in  sections,  the  lower  section 
being  22'  8'  long.  The  upper  sections  are  spliced  to  the 
lower  sections  by  means  of  horizontal  plates,  angle  irons, 
and  splice  plates,  the  joint  ;//  w  being  a  short  distance  above 
the  floorbeams. 


(a) 


(b) 


Fig.  13 


51.  In  the  drawings  there  will  be  noticed  several  conven- 
tionalities. There  are  a  number  of  rivet  holes  shown  black 
in  the   side  views    (the   views    having   the   center    lines  c  d 


STEEL  COLUMNS 


JUNE  25.  1693. 


-oflce  of  copyright,  see  pace 


kND  CONNECTIONS. 


lediately  following  the  title  page. 


JOHN     SMITH.   CLASS    N?  4529. 


§  14  MECHANICAL  DRAWING  35 

and  g-  h)  of  both  designs.  This  indicates  to  the  shop  men 
that  the  rivets  destined  for  these  holes  are  so-called  field 
rivets;  that  is,  they  are  to  be  driven  in  at  the  place  where 
the  structure  is  being  erected  and  after  the  columns  are  in 
place.  The  position  of  the  field  rivets  is  indicated  in  the 
front  views  (the  views  having  the  center  lines  a  //'and  e f) 
by  blackening  in  ihQ  position  of  the  holes.  According  to  the 
rules  of  drawing,  their  position  would  be  shown  by  dotted 
lines,  but  the  conventionalism  of  blackening  in  is  resorted 
to  as  better  calling  the  attention  of  the  workmen  to  the  fact 
that  the  rivets  destined  for  these  holes  are  field  rivets.  This 
common  usage  is  departed  from  only  in  the  case  where  the 
parts  through  which  the  field  rivets  pass  is  hidden  behind 
some  other  parts  of  the  structure.  Thus,  referring  to  the 
front  view  of  Fig.  1,  field  rivets  are  shown  as  passing  through 
the  plate  and  two  angle  irons  at  the  splice,  and  as  neither 
the  plate  nor  the  angle  irons  are  hidden,  their  position  is 
shown  in  black.  In  the  side  view,  however,  the  horizontal 
plate  at  the  splice  is  hidden  behind  the  splice  plate  and  the 
angle  irons  are  hidden  behind  the  splice  plate  and  the  legs  of 
the  channels;  consequently,  the  field  rivet  positions  are 
shown  in  this  view  by  dotted  lines. 

52,  All  the  rivets  that  are  to  be  driven  in  the  shop 
where  the  column  is  made  are  shown  in  position;  in  some 
drawing  rooms,  in  order  to  save  time,  the  rivets  are  omitted, 
their  position  being  merely  indicated  by  two  center  lines. 
In  order  to  show  the  rivets  clearly,  their  heads  are  shaded 
a  little  in  the  manner  shown,  using  the  bow-pen  for  the 
shading. 

53.  The  dimensions  of  the  channels,  I  beams,  and  Z  bars 
are  not  given  in  detail,  this  not  being  necessary  on  a  working 
drawing.  All  that  is  required  is  to  give  the  size  of  the 
channel,  etc.  The  notes  giving  the  sizes  are  read  as  follows: 
On  the  front  view  of  the  left-hand  design  we  find  a  note 
r>-15"-33-22'-7|"  0;  this  means  two  15-inch  channels, 
33    pounds    per  .foot    of    length,    22'  7|"   long.      The    note 

.1/.  E.     V.—g 


36 


MECHANICAL  DRAWING 


S  U 


on  the  front  view  of  the  right-hand  design  "4-6"  X  3|-" 
,-22.7-Z  Bars"  means  that  four  Z  bars  having  a  depth  of 
6  inches,  legs  3^  inches  wide,  and  weighing  22.7  pounds  per 
foot  of  length  are  to  be  used.  The  note  "  2-12"-40-I  Beams  " 
means  that  two  12-inch  I  beams  weighing  40  pounds  per 
foot  of  length  are  to  be  used.  The  note  "4"  X  3^'  X  yV" 
X  llV'  Angle  "  means  that  an  angle  iron  with  legs  measur- 
ing 4  inches  and  3i  inches,  ^  inch  thick,  and  11^  inches 
long  is  to  be  used. 


T 


i,\^" 


■^/' 


e 


54.     In  order  to  aid  the  student  in  drawing  the  various 
parts  the  dimensions  of  which  are   not  fully  given   on  the 

plate,  sectional  views  of 
them,  fully  dimensioned, 
are  given  in  Fig.  14. 

The  section  of  the 
10-inch  I  beam  is 
given  in  Fig.  14  (a), 
of  the  12-inch  I  beam 
in  Fig.  14  (/;),  of 
the  lo-inch  channel  in 
Fig.  14  (c),  and  of  the 
Z  bar  in  Fig.  14  (</). 

In  Fig.  14  (a),  (/;),  and 
(c),  the  dimensions  are 
given  in  decimals,  or  as 
they  appear  in  the  cat- 
alogue of  the  rolling 
mill.  Since  it  is  prac- 
tically impossible  to  lay 
off  these  decimal  dimen- 
sions to  a  scale  of  IV 
=  1  f  t. ,  it  is  recommended 
to  use  the  nearest  six- 
teenths of  an  inch,  taking 

.31  =yV, -40  =  I,  .41  =  1, 


■f.S6' -i 

(d) 


.4"^ 


-iQ 


(c) 


—■^i  -1 


^ 


"1 


"ii  "-'^ 


V-3f- 
(d) 


<to 


li 


.46 


Fig.  14 
,56  =  ^\, 


,66  =  H,   -'J^  =  \h   ■'^^  =  h   -90  =  h 


§  U  MECHANICAL  DRAWINC  37 

55,  All  rivets  in  these  two  designs  are  f  inch  diameter, 
having-  a  head  1^  inches  in  diameter  and  -^^  inch  high.  The 
holes  to  receive  the  rivets  are  generally  punched  j\  inch 
larger  than  the  shank  of  the  rivet  (the  size  of  a  rivet  is 
denoted  by  the  diameter  of  the  shank  before  the  rivet  is 
driven),  so  that  the  rivet  holes  for  f-inch  rivets  are  punched 
J  #  inch  diameter.  As  a  matter  of  course,  the  rivet  when 
driven  fills  the  hole,  so  that  the  drivoi  size  of  a  rivet 
=  nominal  size  +  tV  inch. 

56.  To  draw  the  plate,  begin  by  drawing  the  center 
lines,  locating  ah,  c  d,  ef,  and  g  h  respectively  2y%",  6^", 
10^^",  and  14yV'  from  the  left-hand  border  line.  The  top 
line  K  L  oi  the  floorbeams  is  to  be  located  7^^"  and  the 
bottom  i  j  of  the  bases  if"  above  the  lower  border  line. 
The  floorbeams  are  shown  broken  off  and  should  be  made 
about  the  same  length  in  proportion  to  their  depth  as  they 
appear  on  the  plate. 

The  reference  letters  printed  in  bold-face  italics  are  to  be 
omitted  on  the  drawing  made  by  the  student. 


DRAWING  PLATE,  TITLE:    TURRET-LATHE  TOOLS 

57.  This  plate  represents  a  working  drawing  of  three 
tools  used  in  connection  with  a  turret  lathe,  viz.,  a  rougli- 
ins:  box  tool,  a  liollow  mill  liolder,  and  a  releasing  die 
holder. 

.58.  The  roughing  box  tool  has  a  body  having  a  shank 
to  tit  a  hole  in  the  turret  of  the  machine;  a  tool  post  and 
adjusting  washer  receives  the  turning  tool  (not  shown), 
which  is  clamped  by  means  of  the  tool-post  screw ;  two  back- 
rest jaws,  which  are  clamped  in  position  by  means  of  the 
back-rest  clamping  bolt  and  adjusted  by  the  back-rest  adjust- 
ing studs  and  nuts,  serve  to  steady  the  work  being  turned. 

59,  The  hollow  mill  holder  has  a  shank  to  fit  the  turret 
and  a  head  recessed  to  take  a  hollow  mill,  which  is  held  from 
turning  by  a  flat-pointed  VV"""^*^^^  setscrew. 


38 


MECHANICAL  DRAWING  §  14 


60.  The  releasing  die  holder  is  composed  of  a  sleeve 
and  a  die  holder;  the  latter  is  recessed  to  receive  a  die  for 
thread  cutting,  the  die  being  held  by  four  pointed  i-inch 
setscreAvs  with  fillister  heads.  (The  head  of  the  screws 
marked  "Releasing  Die  Screws"  is  called  a  flllistei-  liead 
by  the  trade.)  The  releasing  die  holder  permits  a  thread  to 
be  cut  up  close  to  a  shoulder  or  to  any  predetermined  length 
in  a  turret  lathe,  and  when  this  length  has  been  reached, 
the  die  is  left  free  to  revolve  with  the  work.  To  back  the 
die  off  the  Avork,  the  lathe  is  reversed  and  the  turret  moved 
backwards.  This  causes  the  y\-inch  pin  h\  the  end  of  the 
shank  of  the  die  holder  to  drop  into  the  recess  at  the  end 
of  the  sleeve,  it  being  guided  into  it  by  the  helical  surface 
shown.  The  die  now  being  held  stationary  and  the  work 
revolving  backwards,  the  die  backs  off  the  work. 

61.  To  begin,  draw  the  center  line  a  I;  oi  the  front  and 
side  view  of  the  box  tool  at  a  distance  of  4f|  inches  from  the 
upper  border  line.  Locate  the  center  c  of  the  end  view  at  a 
distance  of  7f  inches  from  the  left-hand  border  line  and 
draw  the  end  view  from  the  dimensions  given.  To  the 
lower  left  of  the  end  view  is  shown  a  partial  view  of  the 
back-rest  part  of  the  body,  which  is  given  in  order  to  show 
clearly  the  location  and  depth  of  the  slot  intended  to  receive 
the  back-rest  jaws  and  also  in  order  to  shoAV  the  location  of 
the  adjusting  studs  in  reference  to  the  slot.  This  view  is 
taken  in  the  direction  of  the  center  line  c  d,  looking  towards  r, 
and  can  readily  be  drawn  from  the  dimensions  given  and 
projecting  from  the  end  view  parallel  to  the  center  line  c  d. 

63.  The  side  view  can  be  drawn  partially  by  projecting 
over  from  the  end  view  and  partially  from  the  dimensions 
given  on  the  side  view.  The  thickness  of  the  lug  for  the 
back-x-est  jaws  is  given  on  the  view  to  the  lower  left  of  the  end 
view.  The  center  line  ^^  of  the  tool-post  boss  may  be  located 
at  a  distance  of  6  inches  from  the  right-hand  border  line. 

63.  To  draw  the  top  view,  begin  by  locating  the  center 
line  /i  i  at  a  distance  of  2y\  inches  from  the  upper  border  line. 
Locate  the  center  of  the  boss,  which  reference  to  the  end 


/8  rfy'as.  fo  fif  Ted  Pes.' 


Pow/- — - 
hardened  and 
drair/7  t>fue 


-'i  — 


Cne  of  Ms^  Toe 
TOOL  POST 


-20  Thds, 
J  fofitbody 

One  cf  ff!/s,  Tool  Steel. 

ADJUSTING  WASHER. 


Cne  of  Ihis,  Tod  Steel- 
BACK  REST  CLAMPING 
I 


'^q;  oij^ 


One  cf  Ihis , 

Macfi/nery  Steel. 
TOOL  POST. 


-NuH. 


-2i 


2  if  this, 

tifacfi/nery  Steel 
ADJUSTING  NUT 
AND  STUD  FOR 

BACKREST. 


!        II         !i    I  ^ 

/i'ardened  on  lip J 

2  cf  this,  fiJactJinery  Steel . 
BACK  REST  JAWS. 


THE  SCRANTON  TOOL  WORKS 

SCRANTON,  Pa., 


Turret  Lathe  Tools 

For  Xo.  1  Tmret  Lathe 


MATERIAL  AS  SPECIFIED 


Scale  Full  Size.  Dale  Oct.  25. 1901 


Draivn  tylA/IA/.C. 


a7ecke^  fpyJ.A.G. 


Dnawet-  157. 


0 refer  No.  154.. 


PTTT 


LU 


Cne  of  tfiis.  Machinery  Sreel . 
HOLLOiVMILL  HOLDER. 


'8Th'ds. 
i 


J-      ^ 
One  of  tfiis. 
Tool  Steel,  Hardened. 
MILL  HOLDER  SCREiV. 


four  cf  this. 
Tool  Steel,  Harden 
RELEASING  DIE\ 


JUNE  25, 1893. 


For  notice  of  copyright,  see  page  : 


^    F/n/s^  on/y  where  markec/f. 

One  of  this.  Steel,  Casting. 

BODY. 


ROUGHING  BOX  TOOL 


■^/ 


i ^f— 


rFlat. 


Finish  a/I  oii-er. 


^ 


One  of  this,  Mact7inefy  Sfee/. 

Finish  all  oi^er.  SLEEl/E  FOR 

RELEASING  DIE  HOLDER. 


REI/I/S. 


One  of  this,  Machine/y  Steel. 
DIE  HOLDER,  FOR  RELEASING  DIE  HOLDER. 


NO.   1984 


lediately  following  the  title  page. 


JOHN  SMITH,     CLASS    N?  4529. 


§  U  MECHANICAL  DRAWING  39 

view  shows  to  be  at  a  distance  of  1  l  inches  from  the  center 
line  of  the  shank,   by  drawing  the  center  line  j k  to  inter- 
sect eg.     While  all  the  vertical  lines  can  be  projected  up  from 
the  side  view  and  some  of  the  horizontal  lines  can  be  drawn 
from  the  dimensions  given,  the  location  of  other  horizontal 
lines  must  be  determined  from  the  end  view.      Since  it  often 
occurs  in  making  drawings  that  a  third  view  must  be  con- 
structed by  reference  to  one  of  the  other  two  views  from 
which   points  or   lines  cannot   be  projected,    the  operation 
will  be  explained  in  detail.      The  center  line  h  i  of  the  top 
view  is  imagined  to  be  the  top   edge  of  a   plane  at  right 
angles  to  the  paper ;  and  in  the  end  view,  the  front  edge  of 
this  imaginary  plane  is  given  by  the  center  line  I vi  passing 
through   the  center  c  of  the  shank.      Suppose  we  Avanf  to 
locate  the  edge  ii  of  the  end  view  in  the  top  view.     Then, 
Ave  take  the  perpendicular  distance  of  n  from  the  plane  Im 
with  the    dividers  and  lay  it  off  in  the  top  view  at  right 
angles  from  the  plane  h  i  and  upA\>ards,  drawing  the  line  ;/' 
through   the  point   thus  laid  off.      In    case  of  doubt  as  to 
which  side  of  the  imaginary  plane  of  the  view  being  drawn 
a  certain  point  or  line  to  be  transferred  from  a  view  at  right 
angles  to  it   is  located  on,  simply  imagine  the  center  lines 
representing  the  edges   of  the  imaginary  plane  to  be   pro- 
duced   until    they  intersect.      Then,   everything  Avithin  the 
90°  angle  in  one  vieAv  is  Avithin  the  same  angle  in  the  other 
view;    likeAvise,    everything  Avithin   the    270°  angle   in  one 
vicAv  is  Avithin  the  same  angle  in  the  other  vicAV.     Thus,  if 
Ave  imagine  the  lines  // /  and  I  vi  to  be  produced  until  they 
intersect,   the    edge  ;/   Avill   lay  Avithin  the  270°  angle   and 
must,  consequently,  be  laid  off  from  //  /  npivards,  in  order 
to  be  located  in  the  same  angle  in  the  top  view.      With  this 
explanation  of  transferring  points  and  lines  from  one  view 
to  another,  the  student  will  experience  no  difficulty  in  com- 
pleting the  top  view. 

6-1:.  The  body  having  been  draAvn,  draAV  the  tool  post, 
locating  its  center  line  1  ,V  inches  from  the  left-hand  border- 
line.    The  center  of  the  top  view  should  be  about  1  j3^  inches 


40  MECHANICAL  DRAWING  §  U 

and  the  lower  end  of  the  tool  post  of  inches  below  the  upper 
border  line.  It  will  be  noticed  that  the  hole  for  the  tool- 
post  screw  has  no  thread  shown  in  it.  the  information  that 
it  is  threaded  being  conveyed  by  the  note  "  fg "  Tap, 
ISThds."  On  working  drawings  a  thread  is  rarely  shown 
in  a  hole  which  is  to  be  tapped ;  the  general  rule  is  to  draw 
two  lines  a  distance  apart  equal  to  the  outside  diameter  of 
the  thread  and  to  place  a  note  on  the  drawing  stating  the 
size  of  tap  to  be  used.  This  practice  has  been  followed  on 
the  plate  "  Turret-Lathe  Tools." 

65.  The  center  line  of  the  adjusting  washer  may  be 
located  -iy^g  inches  from  the  left-hand  border  line ;  the  center 
of  the  top  view  may  be  1  ^^g.  inches  and  the  lower  end  of  the 
front  view  3f  inches  below  the  upper  border  line.  Locate 
the  center  line  of  the  tool-post  screw  ^  inch  below  the 
upper  border  line  and  the  right-hand  end  of  the  side  view 
8f^  inches  from  the  left-hand  border  line.  The  center  of 
the  end  view  may  be  f  inch  to  the  right  of  the  right-hand 
end.  The  center  line  of  the  back-rest  clamping  bolt  should 
be  drawn  2yg-  inches  below  the  upper  border  line ;  the  right- 
hand  end  of  the  side  view  may  be  placed  T|-  inches  from  the 
left-hand  border  line  and  the  center  of  the  end  view  \^  inch 
from  the  right-hand  end  of  the  bolt.  Locate  the  center  line 
of  the  adjusting  stud  o^^  inches  below  the  upper  border  line 
and  the  left-hand  end  3f  inches  from  the  left-hand  border 
line.  The  adjusting  nut  being  movable  on  the  stud,  it  may 
be  shown  in  any  convenient  position,  say  about  in  the  posi- 
tion shown.  The  top  line  of  the  back-rest  jaw  may  be 
drawn  ^f^  inches  below  the  upper  border  line  and  its  left- 
hand  end  may  be  located  ^  inch  from  the  left-hand  border 
line.  A  distance  of  f  inch  may  be  left  between  the  two 
views  of  the  jaw. 

66.  The  center  line  of  the  hollow  mill  holder  may  be 
located  3f  inches  above  the  lower  border  line  and  the  center 
of  the  end  view  5|^  inches  from  the  left  hand  border  line.  A 
space  of  ^  inch  mav  be  left  between  the  front  view  and  the 
side  view. 


§  U  MECHANICAL  DRAWING  41 

67.  The  center  line  of  the  releasing-die-holder  sleeve 
should  be  located  4|-  inches  above  the  lower  border  line  and 
the  center  of  the  end  view  4:j\  inches  from  the  right-hand 
border  line.  A  space  of  f  inch  should  be  left  between  the 
end  view  and  side  view.  The  right-hand  end  is  a  helical 
surface  having  a  pitch  of  :!■  inch ;  it  can  be  laid  out  by  means 
of  the  method  given  in  Geometrical  Dratving,  laying  off 
helixes  of  \  inch  pitch  and  |  inch  diameter  and  \  inch  pitch 
and  \  inch  diameter.  In  the  machine  shop  this  helix 
would  be  cut  by  gearing  the  lathe  to  cut  a  thread  of  4  to 
the  inch  and  using  a  flat  side  tool  set  with  its  cutting  edge 
at  right  angles  to  the  center  line  of  the  sleeve.  The  center 
line  of  the  die  holder  should  be  located  2  inches  above  the 
lower  border  line  and  the  center  of  the  end  view  6|-  inches 
from  the  right-hand  border  line.  A  space  of  y^  inch  should 
be  left  between  the  end  view  and  the  side  view. 

68.  The  releasing-die  screw  and  mill-holder  screw  should 
have  their  center  lines  drawn  Ifi-  inches  above  the  lower 
border  line.  The  center  of  the  end  view  of  the  mill-holder 
screw  may  be  placed  \\\  inches  and  the  center  of  the  end 
view  of  the  releasing-die  screw  7yV  inches  from  the  left-hand 
border  line.  A  space  of  f  inch  should  be  left  between  the 
end  views  and  side  views  of  the  screws. 

69.  All  the  notes  on  the  drawing,  in  accordance  with 
the  most  general  practice,  are  Avritten  in  capital  and  small 
letters,  the  former  being  ^\  inch  and  the  latter  J^-  inch  high. 
Detailed  instruction  for  this  style  of  lettering  have  been 
given  in  Geometrical  Draiving.  The  names  of  the  different 
parts  are  written  in  capitals  ^  inch  high. 

70.  This  drawing  plate  differs  from  those  previously 
given  in  that  the  title  is  placed  in  the  lower  left-hand  corner. 
Different  drawing  offices  have  different  rules  in  regard  to 
titles  for  a  drawing;  some  insist  on  having  all  titles  placed 
in  the  lower  left-hand  corner  and  others  place  it  in  the  lower 
right-hand  corner;  more  rarely  one  of  the  upper  corners  is 
selected.  In  the  arrangement  of  the  title  and  the  informa- 
tion contained  therein,  practice  also  greatly  varies;  this  plate 


42  MECHANICAL  DRAWING  §  U 

simply  shows  the  practice  of  one  drawing  office,  giving  the 
name  and  place  of  the  firm,  the  title  of  the  drawing,  the  mate- 
rial, the  scale,  the  initials  of  the  persons  who  made  the  drawing 
and  checked  it,  the  date,  the  number  of  the  drawer  in  which 
the  tracing  is  kept,  and  the  number  of  the  drawing.  "When  a 
number  is  given  to  a  drawing,  it  generally,  but  not  always  by 
any  means,  gives  the  total  number  of  drawings  that  have  been 
made  in  a  particular  drawing  office.  Since  practice  varies 
so  much,  it  can  only  be  found  by  inquiry  at  the  office  where 
the  drawing  was  made  what  a  number  on  a  drawing  signifies. 

71.  To  draw  the  title,  construct  a  rectangle  S^  X 
2A-  inches,  placing  it  4-  inch  from  the  left-hand  and  the  lower 
border  line.  Divide  it  vertically  into  four  equal  parts  and 
subdivide  each  of  the  two  lower  parts  into  two  equal  parts 
again;  then  draw  the  horizontal  lines  shown.  Divide  the 
-three  lowest  divisions  horizontally  into  two  equal  parts  and 
draw  the  vertical  line  shown.  The  firm  name  is  printed  in 
a  style  of  letter  very  similar  to  a  block  letter,  the  letters 
being  J':r  inch  high.  The  location  of  the  firm  is  printed  in 
block  letters  ^  inch  high,  excepting  the  first  letter  of  each 
word,  which  is  3%  inch  in  height.  The  line  "  Turret-Lathe 
Tools"  is  a  light-face  block  letter  3^0  inch  high,  except  the 
first  letter  of  each  word,  which  is  y\  inch  high.  The  line 
"  For  No.  1  Turret  Lathe  "  is  printed  in  italics,  the  capitals 
and  numeral  being  4-  inch  high  and  the  small  letters  3^  inch 
high.  The  next  line  is  printed  in  light-face  block  letters 
-A-  inch  hieh,  and  the  succeeding  lines  are  written  free- 
hand  in  the  same  style  used  in  all  the  notes,  the  capitals  and 
numerals  being  ^^  inch  and  the  small  letters  -^  inch  high. 
It  is  optional  with  the  student,  as  far  as  the  first  five  lines 
of  the  title  is  concerned,  whether  to  write  them  in  the  free- 
hand style  used  for  the  last  three  lines  or  in  the  different 
styles  shown  on  the  plate. 

73.  The  number  appearing  in  the  right-hand  lower 
corner  is  to  be  made  -^  inch  high. 

The  reference  letters  printed  in  bold-face  italics  should 
be  omitted  on  the  drawing  made  by  the  student. 


CDMML 


C/ampng  Bolt 
and  Washer. 
8  oftfifs.  ^    I 


Cne  cf /-hi. 
Steel  Cast. 
Spider. 
Scale  3  =/ 


44-—, 


JUNE  25,  1693 


For  notice  of  copyright,  see  page 


FATOR 


16-1  ft. 


/90  of  this. 
Drawn  Copper. 
Commutator  Bar. 
Scale :  half  size, 


mediately  following  the  title  page. 


JOHN      SMITH,     CLASS   N?    4529. 


§  14  MECHANICAL  DRAWING  43 

DRAWING    PLATE,    TITLE:     COMMUTATOR 

73.  This  plate  shows  the  detail  ilrawings  and  an  assem- 
bly drawing  of  a  dynamo  eommutator.  It  is  composed  of  a 
spider  and  clamping  ring,  which  are  drawn  together  by 
eight  f-inch  bolts  and  clamp  190  commutator  bars  sepa- 
rated by  mica  insulation  .04  inch  thick.  The  commutator 
bars  are  insulated  from  the  spider  and  clamping  ring  by 
mica  rings  -jV  inch  thick. 

74.  Begin  by  drawing  the  clamping  ring,  locating  its  hor- 
izontal center  line  4|  inches  below  the  upper  border  line  and 
its  vertical  center  line  2f  inches  from  the  left-hand  border 
line.  Leave  a  space  of  ^|  inch  between  the  two  views.  In 
the  sectional  view,  which  is  a  section  taken  on  the  vertical 
center  line,  it  will  be  noticed  that  the  two  largest  diameters 
are  given  to  ten-thousandths  of  an  incli.  In  a  shop  where 
many  commutators  are  made  from  the  same  draAving,  the 
clamping  surfaces  (those  in  contact  with  the  insulation)  of 
the  clamping  ring,  spider,  and  commutator  bars  are  turned 
to  fit  gauges  prepared  by  the  toolmaker,  who  lays  out  the 
gauges  and  makes  them  to  suit  the  dimensions  given.  AVhile 
it  cannot  be  expected  that  he  will  get  such  large  gauges  as 
are  here  required  correct  to  yoio o  inch,  still  the  giving  of 
the  dimensions  in  such  a  small  subdivision  of  the  inch  calls 
his  attention  to  the  fact  that  great  accuracy  is  required. 
This  practice  of  giving  accurate  dimensions  in  decimals  and 
approximate  dimensions  of  a  machine  part  in  halves,  quar- 
ters, eighths,  sixteenths,  etc.  of  an  inch,  is  now  largely 
adopted  in  the  better  class  of  drafting  rooms,  the  purpose 
of  its  adoption  being  to  show  the  workman  at  a  glance 
which  parts  of  a  machine  part  require  to  be  very  accurate 
and  which  do  not  require  it,  thus  tending  to  prevent  the 
waste  of  time  incidental  to  needless  accuracy. 

75.  The  spider  consists  of  an  outer  shell  and  a  hub 
joined  together  by  eight  arms.  The  hub  has  a  keyway  for 
a  \"  X  f  key,  which  is  let  half  into  the  shaft  and  half  into 
the  hub,  making  the  depth  of  the  keyway  in  the  hub  f  inch. 


U  MECHANICAL  DRAVrlXG  ^  14 

Locate  the  vertical  center  line  in  line  with  the  vertical  cen- 
ter line  of  the  clamping  ring  and  locate  the  horizontal 
center  lire  3^  inches  above  the  lower  border  line.  Draw  this 
horizontal  center  line  clear  across  the  sheet,  as  it  also  serves 
for  the  assembly  drawing.  A  space  of  f  inch  should  be  left 
between  the  end  view  and  the  sectional  view  of  the  spider. 

76,  The  center  line  of  the  clamping  washer  and  clamp- 
ing bolt  should  be  located  7f  inches  from  the  left-hand 
border  line ;  the  top  of  the  head  should  be  "2^  inches  from 
the  upper  border  line.  Owing  to  the  small  scale  to  which 
the  bolt  is  drawn,  it  is  difficult  to  draw  the  correct  number 
of  threads  per  inch  (10),  and  hence  the  thread  is  shown 
exaggerated.  This  remark  also  applies  to  the  tapped  holes 
in  the  clamping  ring. 

77,  Locate  the  bottom  line  of  the  commutator  bar 
4^  inches  below  the  upper  border  line  and  locate  the  right- 
hand  boundary  line  1\^  inches  from  the  right-hand  border 
line.  Locate  the  center  line  of  the  end  view  f  inch  from 
the  right-hand  border  line.  The  commutator  bar  is  dimen- 
sioned with  the  dimensions  required  by  the  toolmaker  for 
making  the  gauges  and  b}'  the  machinist  for  turning  it  to 
size;  consequently,  some  of  the  dimensions  required  for 
drawing  it  must  be  obtained  by  calculation.  Thus,  to 
obtain  the  depth  of  the  bar  at  the  center,  we  have 
14.5  inches  as  the  inside  diameter  and  21  inches  as  the  out- 
side diameter  of  the  ring  of  which  the  bar  is  a  segment. 

21  —  14  5 
Then,  the  depth  of  the  bar  is =  3:^  inches.     Other 

dimensions  are  obtained  in  a  similar  manner.  In  regard  to 
laying  oflE  the  dimensions  given  in  decimals,  those  appearing 
on  the  detail  drawings  made  to  a  scale  of  3'  =  1  ft.  should 
be  laid  off  to  the  nearest  thirty-second  inch  (on  a  scale  of 
3"  =  1  ft.),  while  those  on  the  commutator  bar  should  be  laid 
off  to  the  nearest  sixty-fourth  inch  on  a  scale  of  6"  =  1  ft. 

78,  The  assembly  drawing  is  made  up  from  the  detail 
drawings,  drawing  first  the  spider  in  the  half -sectional  view ; 
then  the  commutator  bar,  the  follower,  and  the  clamping 


JUNE  25./893. 


For  notice  of  copyrigist.  see  paee 


nediately  following  the  title  paire 


s/OffN  SM/Tff,  CLASS  JV9  4529. 


§  14  MECHANICAL  DRAWING  45 

bolt  should  be  drawn.  The  lower  half  of  the  side  view  can 
now  be  drawn.  In  order  to  get  the  assembly  drawing  within 
the  space  available,  only  half  of  the  end  view  is  shown,  it 
being  understood  that  the  omitted  part  is  similar  to  the  part 
shown.  In  a  working  drawing,  in  order  to  save  time,  it  is 
not  customary  to  show  all  the  commutator  bars  on  the 
assembly  drawing,  only  a  few  being  indicated,  as  shown. 
The  insulation  between  the  bars  being  only  .04  inch  thick, 
it  is  not  feasible  to  draw  it  to  a  scale  of  3"  =  1  ft.  and  define 
its  thickness  correctly  by  two  lines;  consequently  the  thick- 
ness is  somewhat  exaggerated  in  the  drawing,  which  under 
the  circumstances  is  a  permissible  liberty.  Part  of  the 
mica  insulation  between  the  commutator  bars  and  the  spider 
and  clamping  ring  projects  beyond  the  end  surfaces  of  the 
bars;  it  is  wrapped  securely  with  a  layer  of  heavy  twine 
well  shellacked.  In  the  assembly  drawing,  the  vertical  cen- 
ter line  of  the  end  view  is  to  be  located  ^  inch  from  the 
right-hand  border  line  and  the  left-hand  surface  of  the 
spider  7yV  inches  from  the  same  border  line. 


DRAWING    PLATE,   TITLE:     SHAFT   HANGER 

79.  This  plate  shows  a  drawing  of  a  lianger  used  to 
support  shafting.  There  are  shown  a  side  view,  half  in  ele- 
vation and  half  in  section  on  the  line  Y Z,  part  of  a  top  view, 
part  of  a  bottom  view,  a  complete  longitudinal  section  on 
the  center  line  in  n  at  right  angles  to  the  plane  of  the  paper, 
and  two  small  sections  through  the  frame.  For  convenience 
in  describing,  the  different  views  will  be  numbered. 

Draw  Fig.  1  first,  which  shows  a  half  elevation  and  a  half 
section.  No  great  difficulty  should  be  experienced  in  draw- 
ing it.  It  will  be  noticed  that  the  journal-box  through 
which  the  shaft  passes  has  a  spherical  bearing;  this  is  shown 
by  the  fact  that  the  radius  of  the  curve  is  the  same,  or 
1^  inches,  in  Fig.  1  and  Fig.  2,  the  latter  being  a  view  at 
right  angles  to  Fig.  1.  The  object  to  be  attained  by  making 
the  bearing  partly  spherical  is  to  provide  for  self-adjustment 
in  any  direction,  in  case  the  shaft  is  not  perfectly  straight. 


46  MECHANICAL  DRAWING  §  li 

Fig.  2  shows  very  plainly  how  the  shaft  ma)-  be  adjusted 
vertically.  The  setscrews  A,  A  are  released,  and  the  large 
screws  B  and  B'  are  screwed  up  or  down,  according  as  the 
shaft  is  to  be  raised  or  lowered,  the  spherical  bearing  allow- 
ing of  adjustment  at  every  point.  The  setscrews  are  then 
screwed  down  or  set,  so  as  to  prevent  B  and  B'  from  chan- 
ging their  positions.  It  Avill  be  noticed  that  the  journal-box 
is  made  in  two  parts.  Fig.  2  is  a  section  on  a  line  ;// ;/  and 
shows  the  whole  length  of  the  hanger.  The  part  sectioned 
at  £  and  E  is  shown  in  Figs.  1  and  4  by  £  £.  The  dotted 
lines  between  them  show  the  projection  of  the  boss  drawn 
at  F,  in  Figs.  1  and  4,  but  not  seen  in  this  view.  The  lines 
forming  the  outline  £  D  £'  £  show  the  projections  of  the 
hanger  frame.  The  screws  B  and  B'  are  made  hollow  to 
lighten  them.  The  dotted  ellipse  in  B  is  the  projection  of 
the  rib  shown  at  L  in  Fig.  1. 

Fig.  3  is  a  bottom  view  as  far  as  the  line  C  C,  after  the 
oil  pan  and  screw  B'  have  been  removed.  D  is  the  projec- 
tion of  the  boss  D  in  Figs.'  1  and  2,  through  which  the  set- 
screw  A  passes. 

Fig.  4  is  a  partial  top  view.  This  can  be  drawn  from  the 
dimensions  given  without  an}-  special  instructions. 

Fig.  5  is  a  section  through  CD,  Fig.  1,  and  G //,  Fig.  2. 
It  shows  the  shape  of  the  frame,  which  is  also  shown  by  the 
dotted  outline  P,  in  Fig.  4.  This  section  is  drawn  more 
particularly  to  show  that  the  radius  of  the  curve  on  each 
side  of  any  section  of  the  vertical  part  of  the  frame  is  the 
same. 

Fig.  6  is  a  section  through  A  B,  Fig.  4,  and  IK,  Fig.  1 . 


BRAWrS  G  PI.ATE,   TITI.E  :    BEXC  H  ^^SE 

80.  This  plate  shows  a  bencli  ^-Ise  and  Its  details. 
A  drawing  of  this  kind  is  called  a  detail  drawrog.  Fig.  1 
is  a  complete  top  view  and  Fig.  2  is  a  section  through  the 
center  line  C£>.  The  remaining  figures  are  drawings  of  the 
different  parts,  or  details.  The  actual  practice  in  the  draw- 
ing room  would  be  to  draw  Figs.  1  and  2  first  and  then  the 


BENC 


-jijr-  ~:r. 


F^3. 


JUHja  BS./S9S 


Fcvr  Boitioe  of  ooipyrigSat,  see  paue  q 


u 


u±iid : 


FiyS 


^2f—^~ 


sdiately  following  the  title  page. 


Zo22gttudi?zal  Section  Throu^h.^lJF_ 


JO-H^ SJi^riT^,  CZASSJ^S  4529. 


§  14  MECHANICAL  DRAWING  47 

details,  but  in  the  present  case  the  student  will  do  well  to 
draw  the  details  first,  as  it  will  help  him  in  drawing  the 
other  two  views,  particularly  since  nearly  all  the  dimen- 
sions are  given  on  the  details.  Following  out  this  plan,  leave 
space  for  Figs.  1  and  2  and  begin  by  drawing  Fig.  3.  This 
consists  of  three  views  of  tlie  jaw,  the  part  marked  yJ  in  Figs.  1 
and  t>.  The  parts  marked  /.',  C,  and  D  in  Figs.  1,  2,  and  3 
are  circular  in  shape,  so  as  to  pern.it  the  back  jaw  A  of  the 
vise  to  swing  when  the  pin  E  is  removed.  This  allows  the 
vise  to  hold  tapered  i)ieces  with  as  much  firmness  as  straight 
ones. 

Fig.  4  is  a  detail,  with  dimensions,  of  the  pin  E. 

Fig.  5  is  a  detail  of  that  part  of  the  vise  marked  E  in 
Figs.  1  and  2,  and  also  of  the  wrought-iron  nut  G.  The 
reason  that  only  a  part  of  the  nut  G  is  sectioned  in  Fig.  2  is 
that  a  conventional  method  has  been  used.  In  reality  the 
whole  of  the  nut  should  be  sectioned.  It  would  be  impos- 
sible to  take  a  section  of  the  nut  in  the  manner  shown. 

Fig.  G  shows  a  top  view,  longitudinal  section,  and  cross- 
section  of  the  front  jaw.  This  is  cored  out  to  admit  the 
screw  and  the  nut.  The  form  of  the  cored  part  is  shown  by 
the  longitudinal  section  and  cross-section  through  E E  and 
A  B.  The  front  jaw  moves  in  and  out  with  the  screw,  as 
shown  by  Fig.  2,  the  collar  L  being  held  in  place  by  the 
small  setscrew.  This  is  sufficient,  since  there  is  no  stress 
on  the  collar  beyond  the  force  required  to  pull  the  jaw  out- 
wards, the  force  exerted  by  screwing  the  jaws  together 
coming  entirely  on  the  surface  M  oi  the  screw  head. 

A  separate  detail  drawing  of  the  screw  is  not  required, 
since  all  the  dimensions  are  given  in  Fig.  2.  The  threads 
of  the  screw  are  shown  the  way  they  really  appear,  but  the 
student  can  draw  them  in  the  conventional  way  explained  in 
Art.  24.  Figs.  1  and  2  can  now  be  drawn,  the  necessary 
dimensions  being  oi)tained  from  the  details. 

For  ordinary  shop  drawings,  the  details  are  usually  drawn 
to  a  larger  scale  than  the  general  drawings.  Figs.  1  and  2. 
In  this  case  the  size  of  the  plate  would  not  permit  an 
enlargement. 


48  MECHANICAL  DRAWING  |  14 

BRAAVIXG    PI.ATE,  TITLE:    PROFILES    OF 
GEAR -TEETH 

81.  If  a  circle  is  rolled  on  a  straight  line  without  sliding, 
a  point  on  the  circumference  of  the  circle  will  describe  a 
curve  called  the  cycloid.  The  circle  is  called  the  genera- 
ting circle.  The  shape  of  the  curve  and  the  manner  of 
drawing  it  are  shown  in  Fig.  1.  Let  O  be  the  center  of  the 
generating  circle,  Avhich  is  If  inches  in  diameter,  P  the 
point  on  the  circumference  of  the  generating  circle,  and 
A  B  the  straight  line  on  which  the  generating  circle  is 
rolled  and  which  is  equal  in  length  to  the  circumference  of 
the  generating  circle,  or  If "  X  3.1416  =  5.4978,  say  5|  inches. 
The  generating  circle  should  be  so  placed  that  its  center  O 
lies  over  the  center  of  the  line  A  B,  as  shown.  Divide  the 
generating  circle  into  any  number  of  equal  parts,  in  this 
case  12,  or  P 1,  1-3,  2-3,  3-Jf,  etc. ,  and  through  these  points 
draw  lines  CD,  E F,  G H,  etc.  parallel  to  the  line  AB. 
Through  the  center  O  of  the  generating  circle  draw  the 
radius  O  6.  Divide  each  half  of  the  line  A  B  into  half  the 
number  of  equal  parts  that  the  generating  circle  is  divided 
into,  as  .^  1,  1-2,  2-3,  etc.,.  and  through  these  points  draw 
lines  perpendicular  to  A  B  terminating  in  the  line  G  H, 
as  A  G,  1-1' ,  2-2',  3-3',  etc.  From  the  point  1',  with  a  radius 
equal  to  the  radius  of  the  generating  circle,  as  06  or  I'-l, 
describe  an  arc  intersecting  the  line  K L  in  the  point  /*' ; 
from  the  point  2',  with  the  same  radius,  intersect  the  line  // 
in  the  point  P^ ;  from  the  point  3',  with  the  same  radius, 
intersect  the  line  G H;  continue  in  a  similar  manner  with 
the  remaining  points  Jf.' ,  5',  7',  8',  etc.,  intersecting  the 
lines  EF  and  CD  in  the  points  P\  P\  P\  P\  etc.  The 
points  A,  P\  P',  P\etc.  are  points  in  the  curve  through 
which  the  cycloid  may  be  drawn.  It  will  be  noticed  that 
when  the  center  O  of  the  generating  circle  coincides  with 
the  point  G,  the  point  P  on  the  circumference  of  the 
generating  circle  coincides  with  the  point  A  ;  and  that  when 
the  generating  circle  is  revolved  towards  the  right,  without 
sliding,  until  the  center  O  coincides  with  the  point  1\  the 
point  PwiW  coincide  with  the  point  P\    Thus  it  is  seen  how 


PRDFILES  DF 


V\\\\\     ///// 


/ 


J^zyJ. 


J^U-^Ea5./693. 


For  notice  ot  copyright,  see  page  ir 


3EAR  TEETH. 


diately  following  the  title  page. 


JOHN  SMITH,  CLASS  JV°45a&. 


^  14  MECHANICAL  DRAWING  49 

the  point  P  passes  through  all  the  points  from  A  to  7?, 
namely,  A^  P\  P\  P\  etc.,  when  started  at  A  and  revolved 
towards  the  right  to  B. 

82.  If  the  generating  circle  is  rolled,  without  sliding, 
on  the  outside  of  the  circumference  of  an  arc  of  a  circle 
supposed  to  be  at  rest,  instead  of  being  rolled  on  a  straight 
line,  the  curve  described  by  a  point  P  of  the  generating 
circle  will  be  an  ei)icycloid. 

The  manner  of  drawing  such  a  curve  is  shown  in  Fig.  2. 
A  B  is  the  arc  upon  which  the  generating  circle  is  rolled,  its 
center  being  at  5  and   its   radius    being   3i   inches.      The 
diameter  of  the  generating  circle  is  in  this  case  the  same  as 
in  Fig.  1,  or  If  inches.      Make  the  lengths  of  the  arcs  6  A 
and  6B  equal  to  half  the  length  of  the  circumference  of  the 
generating  circle,  by  first  calculating  the  length  of  half  the 
circumference    of    the    generating    circle    and    drawing    a 
straight    line    tangent    to  the    arc  AGB   at    6,    making    it 
equal  in  length  to  half  the  circumference  of  the  generating 
circle.      Then  make  the  arc  6  A  equal  to  this  line  by  means 
of  the  approximate  method  given  in  Geometrical  Drinuing. 
Divide  the  arc  A  6B  and  also  the  generating  circle  into  the 
same  number  of  equal  parts,  in  this  case  12,  as  A  1,  1-2,  2-3, 
etc.  and  Pi,  1-2,  2-8,  etc.,  and  draw  radii  from  the  center  S 
to  the  points  of  division  on  the  arc  A6  B.     During  the  revo- 
lution of  the  generating  circle,  the  center  O  will  describe  an 
arc  mOn  concentric  with  the  arc   A6B  and   having  the 
same  number  of  degrees  in  it  as  A  6  B.     Produce  the  radii 
just  drawn  to  the  arc  of  center  positions  ?n  On,  intersecting 
this   arc    in    the    points    7;^  i',  i?',  J',  ^',  etc.       Through    the 
points   of   equal    divisions,    i,  L',  5,  etc.,    of    the    generating 
circle     pass     concentric     arcs     having    the    center    S,    as 
CD,EF,GH,I/,SLn(\KL.      With   the   points   l',2',3',^', 
etc.  as  centers  and  radii  equal  to  the  radius  of  the  genera- 
ting circle  describe  arcs  cutting  the  arcs  K  L,  I  J,  G  H,  etc.  in 
the  points  P\  P\  P\  etc.,  which  are  points  on  the  epicycloid. 

83.     When  the  generating  circle  rolls  on  the  inside  of  the 
arc,   the  curve  described  by  a  point  on  the  circumference  is 


50  MECHANICAL  DRAWING  §  U 

called  a  liypocycloicl.  The  method  of  drawing  it  is  similar  in 
all  respects  to  that  just  given  for  the  epicycloid.  The  student 
should  be  able  to  construct  it  from  the  drawing  without 
further  explanation.  The  diameter  of  the  generating  circle 
is  If  inches,  as  before. 

84.  Suppose  that  a  string  is  wound  on  a  cylinder  and 
that  the  end  of  the  string  is  at  the  point  P  in  Fig.  3.  If 
this  string  is  unwound  from  the  cylinder,  keeping  it  con- 
stantly tight,  the  end  Pwill  describe  a  curve  known  as  the 
involute  of  tlie  circle,  or,  more  simply,  the  involute. 
To  construct  it  geometrically,  let  O  be  the  center  of  the 
given  circle  representing  the  cylinder,  which,  in  Fig.  3,  is 
2i  inches  in  diameter,  and  Pthe  free  end  of  the  string  when 
wound  on  the  cylinder.  Divide  one-half  of  the  given  circle 
representing  the  cylinder  into  any  number  of  equal  parts,  in 
this  case  6,  as  Pi,  1-2,  2—3,  etc.,  and  through  each  of  these 
points  draw  tangents  to  the  circle,  as  P^,  P'^,  P^,  etc.  To 
draw  these  tangents,  first  draw  the  radii  01,  0  2,  03,  etc. 
and  then  draw  the  tangents  1  P\  2  P^,  3P^,  etc.  at  right 
angles  to  them.  By  means  of  the  approximate  method 
given  in  Geometrical  Drawing,  find  the  length  of  the  arc  1  P 
and  make  the  length  of  the  tangent  1  P^  equal  to  this  length; 
of  the  tangent  2  P-  equal  to  twice  this  length ;  of  the  tangent 
3  P^  equal  to  three  times  this  length,  and  so  on.  The  curve 
drawn  through  the  points  P\  P^,  P^,  P\  etc.  will  be  the 
required  involute.  The  use  of  these  curves  will  now  be 
explained. 

85.  On  the  plate  entitled  Spur  Gear- Wheels,  Fig.  1,  is 
shown  one-half  of  two  spur  gear-wlieels  in  mesh.  The 
two  dotted  circles  tangent  to  each  other  at  Pare  concentric 
to  the  centers  of  the  gear-wheels  and  are  called  the  pitcli 
circles.  "  The  diameter  of  any  gear-wheel  is  always  under- 
stood to  be  the  diameter  of  its  pitch  circle  imless  it  is  specified 
as  diametei*  at  root  or  diameter  over  all.  The  length 
of  that  part  of  the  pitch  circle  between  the  centers  of  any  two 
consecutive  teeth  is  called  the  circular  pitcli,  or  simply 
the  pitcli.     Thus,  in  the  last-mentioned  figure,  the  length 


§  14  MECHANICAL  DRAWING  51 

of  the  arc  a  b  is  equal  to  the  pitch  of  either,  gear-wheel. 
When  the  gear-wheels  are  cut  in  a  gear  cutter,  the  width  of 
the  tooth  c  dow  the  pitch  line  is  equal  to  the  space  df\  that 
is,  the  arc  f  ^/ is  equal  to  the  arc  d  f^  and  each  is  equal  to 
half  the  pitch.  When  the  gear-wheels  are  cast,  that  is, 
when  they  are  not  cut  in  a  gear  cutter,  clearance  is  given 
between  the  back  of  one  tooth  and  the  front  of  the  tooth 
following,  to  allow  for  inequalities  in  casting.  This  clear- 
ance, or  backlash,  as  it  is  usually  termed,  is  generally  made 
equal  to  4f,$  of  the  pitch.  This  is  done  by  making  the  thick- 
ness of  the  teeth  c  d  equal  to  .48  of  the  pntch. 

The  part  C  C^  of  the  tooth  that  lies  beyond  the  pitch  circle 
is  called  the  addeiKlum,  and  the  part  C  C^  that  lies  below  it  is 
called  the  root.  The  face  of  the  tooth  is  the  part  C  C^  C  C, 
Fig.  2,  of  the  tooth  above  the  pitch  circle,  extending  the 
whole  width  of  the  tooth.  The  flank  is  the  part  C  C^C"  C, 
Fig.  2,  of  the  tooth  below  the  pitch  circle,  extending  the  whole 
width  of  the  tooth.  The  terms  addendum  and  root  mean 
distances  only,  while  face  and  flank  mean  surfaces. 

The  usual  practice  is  to  make  the  addendum  equal  to  .3  P, 
and  the  root  equal  to  A  P.  P=  the  circular  pitch.  The 
distance  C^  C^  is  called  the  whole  depth  of  the  tooth. 
The  method  of  describing  the  curves  of  teeth  shown  on  the 
plate  entitled  Profiles  of  Gear  Teeth,  Fig.  4,  is  a  convenient 
way  of  drawing  the  cycloidal,  or  double-curved  teeth. 
Cycloidal  teeth  are  constructed  by  making  the  outline  of 
the  face  a  part  of  an  epicycloid  and  the  flanks  a  part  of  the 
hypocycloid,  hence  the  name  double-curved  teeth. 

86.  In  Fig.  4  of  the  plate  entitled  Profiles  of  Gear-Teeth, 
let  A  B  be  part  of  a  pitch  circle  struck  with  a  radius  of,  say, 
5Mnches.  For  convenience  in  drawing  the  tooth,  let  the 
pitch  be  2  inches.  With  6^  as  a  center,  which  is  the  center 
of  the  gear-wheel,  and  a  radius  equal  to  5i  inches  describe 
the  arc  A  B,  part  of  the  pitch  circle.  Through  O  draw  a 
straight  line  OS,  cutting  A  B  in  /'.  Take  the  radius  of  the 
generating  circles  .V  /^and  -S"  /'  ccjual  to  1^  inches  for  this 
case  and  describe   arcs  having  centers  at  i^  and  5' on  the 

M.  E.   y.~ij 


52  MECHANICAL  DRAWING  §  14 

line  O  S.  With  O  as  center  and  (9  5  as  radius  describe 
the  arc  5,  5,.  In  connection  with  the  gear-wheel  teeth,  the 
generating  circles  are  frequently  called  describing  circles. 
Roll  the  outer  describing  circle  upon  A  B  in  such  a  manner 
that  the  center  5  will  move  in  the  direction  of  the  arrow 
along  the  arc  5, 5,.  By  means  of  the  method  given  in  Fig.  2, 
find  the  points  P\  P',  /",  etc.  on  the  epicycloid  described 
by  the  point  P.  Trace  a  faint  curve  through  the  points  just 
found  and  measure  off  on  the   pitch  circle  the  thickness  of 

the  tooth. 

PD  =  .48/  =  .48  X  -r  =  .96'. 

Make  £F=the  addendum  =  . 3  X  /  =  . 3  X  2'  =  . 6'. 

With  O  as  a.  center  and  O  F  sls  a  radius  describe  an  arc 
cutting  the  epicycloid  in  G.  Now  roll  the  inner  describing 
circle  on  A  B,  so  that  its  center  5 '  moves  in  the  direction 
of  the  arrow,  and  find  the  points  P,,  /*„  P^,  etc.  of  the 
hypocycloid  described  by  the  point  /*,  through  which  trace 
a  faint  curve.  Make  EF'  equal  the  flank  of  the  tooth 
=  . 4/  =  .4  X  2'  =  .8',  and  with  6?  as  a  center  and  OF'  as  a 
radius  describe  an  arc  cutting  the  hypocycloid  in  G' . 
PG'  is  the  outline  of  the  flank  of  the  tooth  and  PG  that 
of  the  face.  Since  it  would  be  a  tedious  operation  to  draw 
all  the  tooth  curves  in  this  manner,  it  is  usual  to  approxi- 
mate the  curves  by  means  of  circular  arcs;  that  is,  to  find 
by  trial  a  center  Q  and  a  radius  QP  such  that  an  arc 
described  from  this  center  and  with  this  radius  will  pass 
through  the  points  on  the  curve  G  P  and  coincide  with  that 
curve  as  closely  as  possible ;  also,  to  do  the  same  with  regard 
to  the  curve  PG',  using  the  center  Q'  and  the  radius  Q' P. 
To  find  the  center  Q  or  Q'  of  these  circular  arcs  proceed  as 
follows :  With  P  and  G  as  centers  and  any  radius  describe 
arcs  intersecting  in  C  and  C.  Draw  a  straight  line  through 
C  and  C ;  the  center  O  must  line  on  C  C  to  the  left  of  G  P. 
Try  different  points  1,  2,  3,  4,  etc.  on  this  line  as  centers 
and  IG,  2  G,  etc.  as  radii,  and  see  if  one  of  the  arcs  struck 
with  either  one  of  these  centers  and  radii  will  coincide  with 
the  epicycloidal  curve  G  P.  Make  this  circular  arc  fit  the 
Qurve   for   a   short    distance    beyond    G — as  far  as  P^,  for 


§  14  MECHANICAL  DRAWING  53 

example;  this  will  insure  the  arc  being-  more  nearly  correct. 
This  should  be  clone  in  every  case  when  finding  an  approxi- 
mate radius  of  this  kind.  Continue  in  this  manner  until  the 
point  Q  is  found  such  that  an  arc  struck  with  Q  as  2i  center 
and  Q  G  or  Q  P  as  a  radius  Avill  coincide  as  closely  as  possi- 
ble with  G  P.  If  a  circle  were  drawn  with  O  as  a  center 
and  O  C2  as  a  radius,  the  centers  of  "all  the  circular  arcs  of 
the  faces  of  the  teeth  would  lie  in  this  circle,  and  the  radii 
of  these  arcs  would  be  equal  in  length  to  Q  J\  Hence,  to 
find  the  center  (?,  of  the  arc  I)  H  forming  the  back  of  the 
tooth,  take  D  as  a  center  and  QP  as  a  radius  and  describe  a 
short  arc  cutting,  in  (7,,  the  circle  passing  through  Q. 
Then,  with  O^  as  a  center  and  the  same  radius  describe  the 
arc  D H.  In  a  similar  manner  find  the  center  Q'  and 
describe  7^6"',  also  D  H' .  Instead  of  letting  the  flank  form 
a  sharp  corner  at  the  bottom  of  the  tooth,  as  shown  dotted 
at  G\  it  is  usual  to  put  a  small  fillet  there,  as  shown  by  the 
full  line.  This  makes  the  tooth  stronger  and  less  liable  to 
break  or  to  crack  in  casting.  The  entire  tooth  outline  or 
curve  G PG'  or  H D H'  is  called  the  profile  of  the  tooth. 

87.  A  rack  is  a  part  of  a  gear-wheel  whose  pitch  circle  is 
a  straight  line;  the  tops  of  the  teeth  all  lie  in  the  same  plane.* 
A  portion  of  a  rack  and  one  tooth  are  shown  in  Fig.  5. 
Take  the  pitch  the  same  as  before,  then  the  addendum  and 
root  are  also  the  same,  that  is,  .(3  of  an  inch  and  .8  of  an 
inch.  Take  the  radius  of  the  describing  circles  1|  inches, 
as  before.  It  is  evident  that  the  tooth  profile  will  be  formed 
of  parts  of  cycloids  formed  by  rolling  the  describing  (gener- 
ating) circle  upon  the  pitch  line  A  B.  Draw  a  small  part  of 
the  cycloidal  curves,  as  shown  in  the  figure,  by  the  method 
given  in  Fig.  1;  lay  off  the  addendum  and  root  and  find 
the  approximate  radius  in  the  same  manner  as  in  the  last 
figure.  The  centers  of  the  curves  for  the  faces  and  flanks 
of  all  the  teeth  of  the  rack  will  evidently  lie  on  the  straight 

*  As  the  radius  of  a  circle  is  increased  indefinitely,  any  arc  of  the 
circle  approaches  more  and  more  to  a  straight  line;  and  when  the 
radius  becomes  infinite,  the  arc  becomes  a  straight  line. 


54  MECHANICAL  DRAWING  §  14 

lines  passing   through    Q  and   0\  respectively,  and  parallel 
to  the  pitch  line  A  B. 

88.  In  Fig.  6  is  shown  the  manner  of  drawing  the 
involute,  or  single-curve  tootli.  The  profile  in  this 
case  is  formed  of  a  portion  of  an  involute  curve  and  a  portion 
of  the  radius  of  the  pitch  circle.  The  circle  from  which  the 
involute  is  constructed  is  called,  in  this  case,  the  base 
circle.  To  find  it  draw  the  pitch  circle,  of  which  the 
arc  A  B  is  a  part,  with  a  radius  equal  to  h\  inches  and 
having  its  center  at  O.  Draw  any  radius  O  W  cutting  the 
arc  A  B  in  D.  Through  D  draw  the  straight  line  E  F, 
making  an  angle  of  To'  with  O  U\  With  O  as  a.  center  and 
a  radius  to  be  found  by  trial,  draw  a  circle  tangent  to  £  F. 
This  circle,  of  which  the  arc  H  G  is  a  part,  is  the  base  circle, 
and  cuts  O  W  \n  P.  Upon  this  circle  construct,  in  exactly 
the  same  manner  as  Avas  shown  in  Fig.  3,  a  portion  of  an 
involute  curve,  passing  through  P.  Lay  off  the  addendum 
I K^  .G  inch,  and  with  O  as  a  center  and  O I  as  a  radius 
describe  an  arc  to  form  the  top  of  the  tooth,  intersect- 
ing the  involute  in  L.  That  part  of  the  flank  below  the 
base  circle  is  straight  and  is  a  part  of  the  radius  drawn 
to  the  point  P.  K P  is  the  root.  The  tooth  has  a  fillet 
at  L  and  R\  as  in  cycloidal  teeth.  A  circular  arc  is  passed 
through  the  points  L  and  /*,  coinciding  as  nearly  as  possible 
with  the  involute  curve  LP.  Its  center  Q  is  found  in  the 
same  manner  as  in  Fig.  4.  For  involute  teeth  it  is  onlj^ 
necessary  to  find  the  one  center  0\  the  centers  for  all  the 
remaining  teeth  lie  on  a  circle  having  O  as  a  center  and 
passing  through  O.  To  draw  the  other  side  of  the  tooth, 
lay  off  on  the  pitch  circle  J/A'=  .1^*6  inch,  as  before. 
With  J/as  a  center  and  QX ^^QP as  a  radius  draw  an  arc 
cutting,  at  ^,,  the  circle  passing  through  0\  with  O^  as  a 
center  and  the  same  radius  describe  the  part  P'  R  oi  the 
tooth  profile  above  the  base  circle.  The  part  P'  R  below 
the  base  circle  is  a  part  of  the  radius  OP'. 

89.  In  drawing  any  of  the  curves  previously  described, 
the  greater  the  number  of  parts  into  which   the  describing 


^  1-t  MECHANICAL  DRAWING  55 

or  base  circles  are  divided  the  greater  will  be  the  accuracy- 
obtained.  The  profile  of  the  rack  tooth  used  for  involute 
gears  is  a  straight  line  making  an  angle  of  15°  with  a  line 
drawn  perpendicular  to  the  pitch  line.  Its  construction  is 
shown  in  Fie.  7. 


DEFINITIONS   AND    CALCULATIONS 

90.  When  a  revolving  shaft  transmits  motion  to  another 
shaft  parallel  to  it  by  means  of  gear-  or  tooth-wheels  in  such 
a  manner  that  two  corresponding  points,  one  on  each  gear- 
wheel, always  lie  in  the  same  plane,  the  two  gears  are  called 
spui*  geai'-'svlieels.  When  the  shafts  are  not  parallel, 
but  their  axes  intersect  in  a  point,  as  O  in  the  plate  entitled 
Bevel  Gears,  they  are  called  bevel  geai'-^vheels.  If  two 
bevel  gear-wheels  that  work  together  have  pitch  diameters 
of  the  same  size,  they  are  called  miter  gear- wheels.  From 
what  has  preceded,  it  is  evident  that  the  circular  pitch  mul- 
tiplied by  the  number  of  teeth  equals  the  circumference  of  the 
pitch  circle. 

Let  /  =  circular  pitch  of  gear-wheel; 

;/  =  number  of  teeth  ; 
d  =  pitch  diameter ; 
TT  =  3.141G  {-  is  pronounced  pi). 

Then,  d  =  ^~,  (1.) 

TT 

or,  the  diameter  of  the  pitch  circle  equals  the  circular  pitch 
multiplied  by  the  number  of  teeth  divided  by  S.lJflG. 

or,  the  circular  pitch  equals  the  pitch  diameter  multiplied  by 
3.1410  divided  by  the  number  of  teeth. 

d~  n  s 

n  =  -,  (3.) 

or,  tlie  Jiumbcr  of  teeth  equals  the  pitch  diameter   multiplied 
by  3.1410  divided  by  the  circular  pitch. 


56  MECHANICAL  DRAWING  §  14 

When  constructing  cycloidal  teeth  for  gear-wheels,  the 
diameters  of  the  describing  circles  are  usually  made  equal 
to  one-half  the  diameter  of  the  pitch  circle  of  a  gear-wheel 
having  12  teeth  of  the  same  pitch  as  those  of  the  gear- 
wheel about  to  be  made. 

Let  li'  be  the  diameter  of  the  describing  circle;  then, 

^'^l^X^,  or.r=^.  (-4.) 

Addendum  =  .3/;  root  =  .4/;  thickness  of  teeth  for  cast 
gears  is  .48/,  and  for  cut  gears  ^/. 


DRA^TXG   PI.ATE,   TITLE  :     SPUR    GEAR-WHEELS 

91.  This  plate  shows  the  halves  of  two  cast  gear- 
wheels having  cycloidal  teeth,  which  work  together,  a 
cross-section  of  each  gear  being  also  given.  The  drawing 
is  full  size,  the  wheels  not  being  shown  entire  for  want  of 
room ;  to  have  done  so  it  would  have  been  necessary  to  make 
the  drawing  to  a  reduced  scale.  The  pitch  is  1  inch,  the 
number  of  teeth  in  the  large  gear  is  36,  and  in  the  small 
one  18.  The  pitch  diameter  of  the  large  wheel  is  found  by 
formula  1  to  be 

d  =  ^   ,  .^  ^  =  11.46  inches,  nearly. 
3.1416  •' 

The  pitch  diameter  of  the  small  gear  = 

1  X  IS       .  .^  .     ,  , 

.^  ,  ,^  ,  =  0.  .3  mches,  nearlv. 
3.1416 

The  diameter  of  the  describing  circle  is  found  by  formula 
4  to  be 

d'  = =  1.91  inches. 

3.1416 

For  all  practical  purposes,  the  diameter  of  the  describing 
circle  may  be  taken  to  the  nearest  16th  of  an  inch.  For 
circular   pitches  under  i  inch,  approximate  the  diameter  of 


J'U^E  25.  /393. 


For  notice  of  copjrright,  see  page  : 


SPUR  GEAR  WHEELS. 


Ca^st  Iron^ 


FtclL  Stze. 


lediately  followint^  the  title  page. 


jrO:H-7<r  SJVTITJ-C.  CZ.Ji.S3JV°4529. 


§  U  MECHANICAL  DRAWING  57 

the  describing  circle  to  the  nearest  ;32d  of  an  inch.  To  find 
the  nearest  10th  or  32d,  multiply  the  decimal  part  of  the 
diameter  by  16  or  32  and  take  the  nearest  whole  number  of 
the  product  as  the  number  of  IGths  or  o'lds  that  the  decimal 
represents.  Thus,  in  the  above,  the  decimal  part  of  the 
diameter  is  .91  inch;  .91  X  16  =  14.56.  The  nearest  whole 
number  to  14.56  is  15;  hence,  the  diameter  of  the  describing 
circle  is  1||-  inches.  If  the  diameter  had  been  required  to 
the  nearest  3-2d,  .91  X  32  =  29.12;  29  is  the  nearest  whole 
number;  hence,  the  diameter  would  be  Iff  inches.  In  this 
case,  take  the  diameter  as  1{|-  inches,  approximating  to  the 
nearest  IGth  for  all  circular  pitches  above  ^  of  an  inch. 
The  addendum  will  be  .3  X  1  inch  =  .3  inch;  the  root  =  .4 
X  1  inch  =  .4  inch;  and  the  thickness  of  the  tooth  on  the 
pitch  circle  =  .48  X  1  inch  =  .48  inch. 

Draw  the  line  of  centers  O  O'  between  the  two  axes. 
With  O  as  a  center  describe  a  semicircle  having  a  diameter 
of  11.46  inches,  cutting  0  0'  in  P.  With  O'  as  a  center 
describe  a  semicircle  having  a  diameter  of  5.73  inches 
which  shall  be  tangent  to  the  first  circle;  this  semicircle 
also  cuts  O  O'  in  P.     These  are  the  pitch  circles.      Divide  the 

larger  pitch  circle  into  — ,   or  18  equal  parts  by  using  the 

protractor.  This  is  accomplished  by  laying  the  protractor 
on  the  drawing  in  such  a  manner  that  the  center  of  the  pro- 
tractor coincides  Avith  the  center  O  of  the  gear-wheel  and 
then  laying  off  on  the  drawing  18  divisions,  each  equal 
to  10°.      The  reason  for  this  Avill  be  clear  when  it  is  remem- 

3('0° 
bered  that  there  are  360°  in  a  circle,  or  -— -  =  180°  in  a  semi- 

z 

circle,  and  as  there  are  18  teeth  in  the  semicircle,  180°  -^  18 
=  10°,  which  is  the  angle  between  two  lines  drawn  from  the 
centers  of  any  two  consecutive  teeth  to  the  center  O.  In  a 
like  manner,  any  circle  may  be  divided  into  parts  by  using 
the  protractor.  Make  C  C^  =  .3  inch  =:  addendum,  and  C  C^ 
=  .4  inch  =  root.  With  O  as  a.  center  and  O  C^  and  O  C^ 
as  radii,  describe  light  circles,  called  acldenduni  and  root 
circles,  to  represent    the  tops  and  bottoms  of  the    teeth. 


58  MECHANICAL  DRAWING  §  14 

Consider  the  points  of  division  just  laid  off  on  the  pitch 
circle  as  the  centers  of  the  teeth,  and  lay  off  on  each  side 
one-half  of  the  thickness  c  d  of  the  tooth,  or  .48  X  | 
=  .'24  inch.  Upon  another  sheet  of  paper  strike  a  short 
arc  of  the  pitch  circle  and  construct  the  profile  of  the  tooth 
as  described  in  the  plate  entitled  Profiles  of  Gear-Teeth,  using 
describing  circles  1|-|  inches  in  diameter.  Having  found 
the  centers  0  and  Q^  of  the  circular  arcs  used  for  the  profiles 
of  the  teeth,  draw  circular  arcs  through  these  centers,  as  pre- 
viously described ;  then,  with  O  (see  plate  entitled  Spur  Gear- 
Wheels)  as  a  center  and  the  same  radii  describe  circles ;  these 
circles  will  contain  the  centers  of  the  circular  arcs  which 
form  the  teeth  profiles.  With  the  point  A  as  a  center  and 
a  radius  equal  to  the  radius  of  the  face  of  the  tooth  (found 
on  the  other  sheet  of  paper  before  mentioned),  describe  an 
arc  intersecting  the  circle  of  face  centers  at  O.  With  0  as 
a  center  and  the  same  radius,  describe  the  arc  A  D  for  the 
face  of  the  tooth,  the  point  D  being  the  point  of  intersection 
of  A  D  with  the  addendum  circle.  In  the  same  way  draw 
the  remaining  faces  of  the  teeth.  To  draw  the  flanks  take 
a  point  representing  the  intersection  of  a  tooth  profile  with 
the  pitch  circle  (the  point  B,  for  example)  as  a  center  and 
a  radius  equal  to  the  radius  of  the  flank  found  on  the  other 
paper  on  which  the  tooth  profile  was  drawn,  and  describe  an 
arc  intersecting  the  circle  of  the  flank  centers  in  O^.  With  O^ 
as  a  center  and  the  same  radius,  describe  the  flank  curve, 
stopping  at  the  root  circle.  Draw  the  flanks  of  the  remain- 
ing teeth  in  the  same  way  and  then  put  in  the  fillets.  The 
remaining  part  of  Fig.  1  can  be  easily  drawn  from  the 
dimensions  given. 

Fig.  2  is  a  conventional  method  of  drawing  cross-sections 
of  gears.  The  hubs  and  rims  are  sectioned,  but  the  teeth 
and  arms  are  not.  This  is  similar  to  the  wheel  shown  on  the 
plate  entitled  Details.  This  method  of  sectioning  makes 
the  views  clearer  and  saves  the  time  spent  in  sectioning. 

In  Fig.  3  the  entire  gear  is  sectioned,  except  the  teeth. 
The  student  should  now  be  able  to  finish  the  plate  without 
further  instructions. 


J^ttch.  cZz.oc.jS.S!' 
Pitch /' 
£0  teeih,. 


JUNE  25.  /693. 


For  notice  of  cop>Tight,  see  page 


BEVEL  GEARS. 

Cexsilro-n       Pzi.ll  Size. 


^•^■'^4^ 


ediately  following  the  title  page. 


i/ay>v  s/if/r/f,  class  n9  4.529. 


§  U  MECHANICAL  DRAWING  59 

DRAWING   PLATP:,   TITLE:    BEVEIi   GEARS 

93.  To  draw  in  section  and  projection  two  cast  bevel 
geai"s  whose  axes  intersect  at  right  angles:  The  number  of 
teeth  in  the  large  gear  is  20,  in  the  pinion  10.  The  circular 
pitch  is  1  inch ;  the  teeth  are  to  be  of  the  cycloidal  form,  hav- 
ing a  face  2  inches  wide.  In  any  kind  of  gearing,  whether 
spur,  bevel,  or  spiral,  the  smaller  wheel  is  called  the  pinion. 

Calculate  the  pitch  diameters,  addenda,  roots,  and  descri- 
bing circles  by  the  same  rules  that  were  given  for  spur  gears. 

r^-         ^         f     ■    ■  16  X  1  inch  , 

Diameter  oi  pmion  =  — — -— — -_ —  =  5.09  . 


Diameter  of  the  large  gear  =  —  —  —  0.37". 


3.1410 

20  X  1  inch 

3.1410 

0x1  inch 

Diameter  of  describing  circle  =    —  =  1.91" 

3.1410 

Take  this  as  l\f  inches,  as  in  the  last  plate. 

Addendum  =  .3  inch;  root  =  .4  inch.  The  sectional  view 
must  be  drawn  first.  'Draw  PP'  and  through  some  point  P 
on  this  line  draw  P'P^  perpendicular  to  it.  Lay  off  PP' 
equal  to  the  diameter  of  the  pinion  =  5.09  inches;  also  P' P^ 
equal  to  the  diameter  of  large  gear  =  0.37  inches.  Bisect 
Py-"  and  /"/*,,  and  draw  6^  J/ and  ON  perpendicular  to 
those  lines  at  the  point  of  bisection;  they  intersect  in  O. 
(9 J/ and  ON  are  the  axes  of  the  two  gears  and  intersect 
at  right  angles  as  required.  Draw  POP^  and  P' O. 
Through  /'draw. -J  /'.]/ perpendicular  to  OP.  Through/" 
draw  M P' N  perpendicular  to  OP'  and  through  P^ 
draw  /' .V  perpendicular  to  O  P^.  PJ/and  P'  M  intersect 
at  J/  on  the  line  C^J/;  P'  N  and  /',  iV  intersect  at  .V,  on 
the  line  ON.  Lay  off  /''  C,  PC,  and  P,  C\,  each  equal  to 
2  inches,  or  the  width  of  the  face  of  the  teeth ;  these  lines 
are  called  the  pitch  lines,  and  the  width  of  the  face  of  the 
teeth  is  always  measured  on  these  lines.  Lay  off  PA  equal 
to  .3  inch  =  the  addendum,  and  /'/>  equal  to  .4  inch  =  the 
root.  Lay  off  P'  E  and  /"  D  for  the  addendum  and  root  of 
the  other  side,  and  P'  P.'  and  P'  D^  for  the  addendum  and 


60  MECHANICAL  DRAWING  §  U 

root  of  the  large  gear.  All  these  addenda  and  roots  are 
each  equal  to  .3  inch  and  .4  inch,  respectively.  In  bevel 
gears,  all  straight  lines  of  the  tooth  profiles  pass  through 
the  point  of  intersection  O  of  the  axes;  hence,  draw  ^-J  (9, 
and  A  A  will  be  the  projection  of  the  top  of  the  tooth. 
Draw  B  O.  and  B  B '  will  represent  the  bottom  of  the  tooth, 
the  line  A'  C B'  being  perpendicular  to  OP.  Make  BF\ 
D F,  D^  F^,  etc.  each  equal  to  \  inch,  according  to  dimen- 
sions. Join  F' ,  F,  F^,  and  F^  with  O,  intersecting  the  per- 
pendiculars through  C,  C,  and  C^  (namely,  the  lines  .-J  '  C  B\ 
etc.  produced)  at  G',  G,  6",,  and  G„.  G'  G  and  G^  6^„  will 
represent  the  bottom  of  the  gears.  The  rest  of  the  sec- 
tional part  can  be  drawn  from  the  dimensions. 

93.  To  show  the  shape  of  the  teeth,  proceed  as  follows: 
For  the  large  gear,  take  A'  as  a  center,  XP'  as  a  radius, 
and  describe  an  arc.  Choose  a  point  H  and  lay  off  H H' 
=  .48  times  the  pitch  =  .48  inch,  or  the  width  of  the  tooth. 
With  XF'  and  XD^  as  radii,  describe  the  addendum  and 
root  circles.  Roll  the  describing  circles  upon  the  arc  whose 
radius  is  XP'  and  construct  the  tooth  profile  in  exactly 
the  same  manner  as  in  Fig.  4  of  the  plate  entitled  Profiles 
of  Gear-Teeth,  O  H  and  0^1/'  being  the  radii  of  the  faces 
and  flanks.  *  To  show  the  shape  of  the  same  tooth  at  C, 
draw  C X  perpendicular  to  OP',  or,  what  is  the  same  thing, 
parallel  to  XP'.  With  X'  C  as  a  radius  and  X  as  a  cen- 
ter, describe  an  arc.  Draw  A'//"  and  X H',  and  the  distance 
between  the  points  of  intersection  on  the  arc  just  drawn, 
measured  on  that  arc,  will  be  the  pitch  of  the  gear  at  the 
bottom  of  the  tooth.  With  the  same  center  and  A  '  E^ 
and  X'  E,  as  radii,  describe  arcs  representing  the  addendum 
and  root  circles.  Draw  A'^^and  X Q^,  also  (2 //and  Q^H'. 
Through  K  draw  K Q  parallel  to  HO,  and  through  K' 
draw  K'  O^  parallel  to  H'  0^\  the  points  of  intersection  Q 
and  Q^  of  these  lines  with  A'  Q  and  X  Q^  are  the  centers  for 
the  face  and  flank  of  the  tooth  at  K  and  K' .  Circles  pass- 
ing through  these  points  concentric  with  A'  contain  the 
centers  of  all  the  circular   arcs  forming  the  tooth  profiles 


§  14  MECHANICAL  DRAWING  61 

that  may  be  laid  off  upon  the  arc  whose  radius  is  N K. 
The  whole  process  is  called  developing  the  teeth  of 
bevel  goal's. 

In  the  same  manner  construct  the  tooth  curves  for  the 
pinion,  using-  the  same  describing  circles,  Iff  inches  in  diam- 
eter, and  MP',  J/'  C  as  radii,  instead  of  N P'  and  N'  C . 

9-4,  To  construct  the  other  view,  draw  first  the  projec- 
tion of  the  pinion.  Draw  the  center  line  ///  //.  Produce  the 
lines  F F\  D  B,  P'  P,  and  E  A  across  the  drawing,  as  shown. 
Choose  a  point  5  on  ;;/ ;/  as  a  center  and  draw  a  quadrant 
with  a  radius  equal  to  the  radius  of  the  pinion,  as  S  P. 
Project  the  points  D  and  E  upon  MO  in  D^  and  E^.  "With  5 
as  a  center  and  the  distances  E^  £"and  D^  D  as  radii,  describe 
quadrants  to  represent  the  tops  and  bottoms  of  the  teeth, 
that  is,  the  projection  of  the  addendum  and  root  circles  of 
the  pinion  in  Fig.  2.  Since  the  whole  pinion  contains 
16  teeth,  the  quadrant  will  contain  4  teeth;  hence,  divide 
the  quadrant  into  4  equal  parts  on  the  pitch  circle  to  repre- 
sent the  centers  of  the  teeth.  Lay  off  on  each  side  of  the 
points  of  division  distances  ge  and  gb^  each  equal  to  one- 
half  the  thickness  of  the  tooth.  On  each  side  of  the  points 
of  division  on  the  addendum  circle  lay  off  Jif  and  //  r,  each 
equal  to  one-half  the  thickness  of  the  top  of  the  tooth  J K, 
Fig.  1,  measured  on  the  addendum  circle.  On  each  side  of 
the  points  of  division  on  the  root  circle  lay  off  id  and  ia, 
each  equal  to  one-half  the  thickness  of  the  tooth  at  the  root, 
as  OP,  Fig.  1,  measured  on  the  root  circle.  Having  now 
three  points  on  each  side  of  all  the  teeth  to  the  right  of  the 
center  line  ;//;/,  project  them  upon  the  lines  EA,  P' P, 
and  D B,  produced  as  shown.  For  example,  project /"and  r 
upon  E  A  in  /'  and  c' ;  e  and  b  upon  P'  P  in  c'  and  //  ;  d  and  a 
upon  DB  in  d'  and  a'.  Draw  a  curve  through  these 
points,  either  by  using  an  irregular  curve  or  by  circular  arcs. 
This  remark  also  applies  to  the  other  curves  shown  in  the 
quadrant. 

95.  The  tooth  curves  in  Fig.  1  must  be  drawn  as  accu- 
rately as  possible,  but  those  shown  in  Fig.  2,  being  oblique 


62  MECHANICAL  DRAWING  §  14 

projections,  are  drawn  to  satisfy  the  eye,  and  no  particular 
accuracy  is  required.  To  find  the  points  on  the  tooth 
curves  at  the  bottom  of  the  pinion,  describe  a  circle  having 
a  center  O,  upon  /////,  which  shaU  be  tangent  to  PP'  and 
have  a  diameter  equal  to  6.37'  =  the  diameter  of  the  large 
gear.  Through  B'  and  A\  Fig.  1,  draw  lines  parallel 
to  O  O,;  also  draw  other  lines  through  (9,  and  the  points  d\ 
f\  c\  etc.,  cutting  the  lines  first  drawn  in  d'\  f",  c",  etc. 
Two  points  are  considered  enough  in  this  case,  as  the  curves 
are  very  short.  They  may  be  drawn  in  with  the  irregular 
curve  in  the  same  manner  as  the  tops.  The  other  teeth  are 
drawn  in  a  similar  manner.  Draw  the  middle  tooth  first. 
The  left-hand  half  of  the  pinion  is  exactly  the  same  as  the 
right-hand  half. 

96.  To  draw  the  projection  of  the  large  gear,  project 
the  points  E' ,  D^,  L,  and  R  upon  the  axis  O  X,  in  the 
points  E^,  Z>j,  Zj,  and  R^,  and  with  O^  as  a  center  and  radii 
equal  to  E^E\  D^D^,  E^L,  and  R^R,  describe  circles  to 
represent  the  addendum  and  root  circles  of  the  tops  and  bot- 
toms of  the  teeth  in  Fig.  2.  Divide  the  pitch  circle  into 
20  equal  parts,  to  correspond  with  the  number  of  teeth  in 
the  large  gear,  beginning  with  the  point  of  intersection  of 
the  pitch  circle  with  the  center  line  ///  n.  Lay  off  on  each 
side  of  these  pitch-circle  divisions,  distances  equal  to  one- 
half  the  thickness  of  the  teeth  =  one-half  of  H H'  in  Fig.  1. 
By  exactly  the  same  method  that  was  used  to  lay  off  the 
thickness  of  the  teeth  at  the  top  and  bottom  on  the  quad- 
rant, lay  off  the  thickness  of  the  top  and  bottom  of  the 
teeth  on  the  addendum  and  root  circles  in  Fig.  2.  Draw 
the  bottoms  of  the  teeth  in  exactly  the  same  manner  as  the 
bottoms  of  the  pinion  teeth  were  drawn. 

All  the  teeth  of  the.  large  gear  are  alike  in  the  projected 
view. 

97.  Bevel  gears  are  always  measured  according  to  their 
largest  pitch  diameter,  as  P P'  and  P'  P^.  If  a  bevel  gear 
were  spoken  of  as  12  inches  in  diameter,  it  would  be  under- 
stood that  the  largest  pitch  diameter  was  12  inches. 


§  U  MECHANICAL  DRAWING  63 

DRAWTXO    PLATE,   TITl^K :    1?RISII    IIOr.DER 

98.  This  drawing  plate  is  a  complete  working  drawing 
of  the  left-hand  brush  holder  for  a  15-horsepower  motor  and 
is  designed  for  the  use  of  carbon  brushes,  there  being  four 
brushes,  two  in  each  holder.  The  carbon  brushes  a,  a  are 
clamped  by  means  of  the  setscrews  /',  /;  to  the  clamps  r,  r, 
which  are  free  to  slide  in  rectangular  holes  in  the  body  d  of 
the  brush  holder.  The  setscrews  b,  b  do  not  bear  directly 
against  the  brushes,  but  against  a  brass  shoe  c.  A  thorough 
electric  connection  between  the  carbon  brushes  and  the  body 
of  the  holder  is  insured  by  flexible  No.  12  cables/,  /",  which 
are  composed  of  strands  of  copper  wire  covered  with  insula- 
ting material.  The  outside  diameter  of  these  cables  is 
-jV  inch,  about.  The  carbon  brushes  are  held  against  the 
commutator  by  hammers  g,  g  operated  by  springs  //,  //. 
The  hammers  are  pivoted  to  brackets  d'  cast  in  one  Avith  the 
body,  and  the  springs  are  so  hung  that  when  the  hammers  are 
rotated  away  from  the  brushes,  the  springs  will  come  to  the 
other  side  of  the  center  around  which  the  hammers  turn 
and  thus  hold  the  hammers  away  from  the  brushes.  The 
springs  are  hooked  over  lugs  on  the  body  at  one  end  and 
over  arms  projecting  from  the  hammers  on  their  other  end. 
In  order  to  insulate  the  brush  holder  from  the  frame  of 
the  machine,  it  is  fastened  to  a  piece  of  hardwood  i  by 
two  3^^-inch  rivets  and  one  No.  10  wood  screw  -J  inch  long. 
The  piece  of  hardwood  is  fastened  to  the  frame  k  of  the 
machine  by  a  1-inch  capscrew  as  shown. 

99.  Begin  dr;iwing  the  plate  by  drawing  the  end  view, 
locating  the  horizontal  center  line  passing  through  the  center 
of  the  rivet  serving  as  a  fulcrum  for  the  hammer  at  a  dis- 
tance of  3j-V  inches  below  the  upper  border  line  and  locating 
the  vertical  center  line  2^-  inches  from  the  right-hand  border 
line.  The  end  view  is  to  be  drawn  first  because  it  is  the 
only  view  in  this  particular  instance  in  which  everything 
can  be  drawn  without  having  to  project  from  another  view. 
For  several  of  the  dimensions  it  is  necessary  to  refer  to  the 
top  view.      The  main   ])art   of   the   hammer  is   flat  and  has 


64  MECHANICAL  DRAWING  §  14 

joined  to  it  a  handle  having  a  circular  cross-section.  The 
flat  and  round  part  coming  together  cause  the  intersection 
curve  shown  at  /.  The  distance  that  the  flexible  cord  / 
projects  from  the  left-hand  face  of  the  body  is  not  given  on 
the  drawing,  as  this  information  would  be  useless  on  a  work- 
ing drawing.  For  the  information  of  the  student  it  is  here 
given,  being  If  inches.  It  is  a  general  and  a  good  rule  with 
draftsmen  not  to  give  any  dimensions  on  a  drawing  unless 
they  serve  a  useful  purpose ;  everything  superfluous  is  to  be 
left  off.  The  shoe  e  is  marked  No.  18  sheet  brass;  the 
corresponding  thickness  is  .04  inch,  nearly. 

It  will  be  observed  that  the  spiral  spring  Ji  is  not  drawn 
the  way  it  actually  appears,  but  that  it  is  drawn  convention- 
ally. This  is  merely  done  in  order  to  save  the  draftsman's 
time,  as  the  note  "  Spiral  Spring  Piano  Wire,  No.  28,  f"  ^, " 
appearing  in  the  top  view  supplies  the  necessary  information 
to  the  mechanic. 

100.  The  end  view  having  been  completed,  the  top 
view  should  be  drawn  next,  locating  the  center  of  the  |-inch 
capscrew  fastening  the  hardwood  strip  i  to  the  frame  S^^g- 
inches  from  the  left-hand  border  line.  This  view  is  drawn 
partially  from  the  dimensions  given  and  partially  by  pro- 
jecting over  from  the  end  view.  The  springs  Ji  being  at  an 
inclination,  they  appear  foreshortened  in  the  top  view,  and 
their  length  in  the  top  view  must  be  determined  by  project- 
ing over  from  the  end  view.  Each  spring  is  hooked  over  a 
horn  cast  on  the  body  d^  a  cylindrical  ring  being  formed  on 
the  ends  of  the  springs  for  this  purpose.  Owing  to  this 
ring  being  at  an  inclination,  it  will  show  elliptical  in  the 
top  view.  The  outside  diameter  of  this  ring  is  -^  inch.  The 
heads  of  the  ■^^"  X  If"  rivets  have  a  radius  of  -3%  inch  and 
are  \  inch  high.  The  head  of  the  No.  10-24  round-headed 
machine  screws  is  |i  inch  diameter  and  -^^  inch  high;  the 
diameter  of  a  No.  10  machine  screw  is  .189  inch,  say  -^-^  inch. 
The  head  of  the  No.  8-32  round-headed  machine  screws 
is  y5_  inch  diameter  and  -^^  inch  high;  the  diameter  is 
.103    inch,    say    W    inch.       ^^lachine    screws    are    made    in 


§  14  MECHANICAL  DRAWING 


65 


accordance  with  the  standard  American  screw  gauge  adopted 
by  all  manufacturers  of  screws;  for  this  reason  it  is  only  nec- 
essary on  a  drawing  to  specify  the  gauge  number  of  the 
screw,  the  number  of  threads  per  inch,  and  the  length,  which 
latter  is  always  measured  under  the  head,  except  in  flat- 
headed  screws,  where  the  length  is  measured  over  all. 
Wood  screws  are  measured  by  the  same  gauge  as  machine 
screws  and  bear  the  same  number.  The  heads  of  the  No.  10 
wood  screws  are  |l  inch  diameter.  The  diameter  across  the 
flats  of  the  head  of  the  J-inch  capscrew  is  f| inch;  the  height 
of  the  head  is  4  inch.  The  ^Vinch  setscrews  have  a  head 
tV  inch  high  and  measuring  j\  inch  across  the  corners. 

101.     To    draw    the    side    view,    begin    by    locating   the 
edge  ;//.^  which  corresponds  to  the  surface  ;//  in  the  end  view, 
at  a  distance  of  1^  inches  above  the  lower  border  line.     The 
view  is  to  be  drawn  partially  by  projecting  doAvn  from  the 
top  view  and  partially  by  transferring  measurements  from 
the  end  view,  as  explained  in  connection  with    the  drawing 
plate,  title:    Turret-Lathe  Tools.      In  order  to  have  a  line 
in  the  end  view  from  which  to  measure,  the  line  represent- 
ing the  surface  ;//  should  be  produced  in  lead  pencil ;  in  the 
side  view  the  edge  ;u  will  then  represent  an  edge  of  the  same 
plane   defined  in  the  end  view  by  the  lead-pencil  line  just 
drawn.      The  shading  of  the  flexible  cords  is  done  freehand. 
102.     The  center  line  of  the  brass  pin   is  to  be  located 
H  inches  from  the  right-hand  borderline;  the  center  of  the 
top  view  is  to  be  placed  5j\  inches  above  the  lower  border  line 
and  a  distance  of  j\  inch  is  to  be  left  between  the  two  views. 
10,3.     The  first  line  of  the  title  is  a  block  letter  j\  inch 
high,  the  second  line  is  ^-V  inch  high,  and  the   third  j\  inch 
high.     The  fourth  and  fifth  line  is  written  freehand,  the 
capitals  being  j\  inch  and  the  small  letters  ^  inch  high.    '  The 
sixth  line  is  composed  of  capitals  j\  inch  high,  except  the  first 
letter  of  each  word,  which  is  |  inch  high.     The  seventh  line 
is  composed  of  capitals  i  inch  high  and  the  eighth  line  is  the 
same   as  the   fourth  and   fifth.      The   student  is  advised   to 
practice  the   block  lettering   shown   in   this  title,    but  if  he 


66  MECHANICAL  DRAWING  §  U 

desires  it  he  may  substitute  the  freehand  letter  shown  in  the 
fourth  line. 

104:.  In  the  upper  left-hand  corner  will  be  found  a  list 
of  the  different  parts  needed.  It  is  the  practice  in  many 
shops  to  have  such  a  list  on  the  drawing  for  convenience  of 
reference.  Draw  the  bottom  line  of  the  list  4^  inches  below 
the  upper  border  line  and  draw  the  upper  line  3f  inches  above 
the  bottom  line  and  divide  the  space  between  the  two  into 
17  equal  spaces.  Draw  the  left-hand  line  of  the  list  ^  inch 
from  the  left-hand  border  line  and  the  right-hand  line 
4 y- inches  from  the  border  line.  To  the  right  of  the  left- 
hand  border  line  of  the  list  draw  vertical  lines  at  a  distance 
of  ^  inch,  -|-  inch,  dj\  inches,  and.  3f  inches,  respectively. 
Then  write  in  freehand,  in  the  style  of  lettering  shown,  all 
the  contents  of  the  list. 

The  student  must  omit  on  his  draAving  the  reference 
letters  printed  in  bold-face  italics. 


READING    A    AVORKIXG    DRAWIJs^G 

105,  The  following  general  method  of  procedure  has, 
by  experience,  been  shown  to  be  conducive  to  the  accurate 
and  rapid  reading  of  a  drawing  made  in  projection.  First, 
if  the  drawing  is  dimensioned,  ignore  the  existence  of  the 
dimension  lines  and  dimensions  entirely  until  after  the 
general  shape  of  the  object  is  fixed  on  the  mind.  Second,  by 
referring  to  the  several  views,  form  an  idea  of  the  shape  of 
the  main  body  of  the  object ;  that  is,  observe  if  its  outline 
shows  it  to  be  a  cube,  a  sphere,  a  cylinder,  a  cone,  a  pyramid, 
etc.,  or  a  combination  of  several  of  these  elementary  forms. 
The  shape  of  the  main  body  having  been  impressed  on  the 
mind,  observe  how  it  is  modified  by  details,  determining,  by 
reference  to  the  several  views,  whether  they  project  from 
the  main  body  or  are  recesses,  or  holes.  Finally,  by  refer- 
ring to  the  dimensions,  form  an  idea  of  the  relative  sizes  of 
the  component  parts.  Pay  due  regard  to  all  conventional 
representations  that  may  have  been   used;   for  instance,  do 


§  U  MECHANICAL  DRAWING  67 

not  become  confused  if  the  ^m  of  a  pulley,  or  a  rib,  which, 
truly  speaking,  should  have  been  in  section,  is  shown  in 
full.  If  two  half  sections  are  placed  on  either  side  of  a 
common  center  line,  remember  that  each  half  must  usually 
be  viewed  independently  of  the  other  and  must  be  mentally 
completed. 

106.  When  reading  a  drawing  in  which  the  views  are 
correctly  placed,  it  is  often  a  great  aid  to  project  points  or 
edges  of  some  part  the  shape  of  which  is  doubtful  over  to 
another  view  by  the  aid  of  a  straightedge,  in  order  to  find 
the  location  of  the  doubtful  part  in  another  view.  When 
the  views  are  not  placed  in  their  correct  relative  positions, 
this  cannot  be  done.  An  example  of  a  case  of  this  kind  is 
given  in  the  plate  Compound  Rest,  and  in  reading  a 
drawing  with  the  view  thus  placed,  the  reader  is  supposed 
to  constantly  imagine  that  the  views  are  in  their  correct 
relative  positions;  with  a  little  practice  this  will  be  found  to 
be  quite  easy. 

107.  In  a  case  of  this  kind,  it  is  manifestly  impossible 
to  project  points  or  lines  from  one  view  to  the  other  by 
means  of  a  straightedge,  and  a  different  method  must  be 
followed.  Select  some  surface  whose  projection  appears  in 
both  views,  or  a  center  line;  now  place  a  pair  of  dividers  so 
that  one  point  rests  on  the  projection  of  the  surface  or 
center  line  selected,  and  open  them  until  the  other  point 
reaches  the  point  or  line  whose  projection  it  is  desired  to 
find  in  the  other  view.  Then  place  one  point  of  the  dividers 
on  the  line  representing  the  selected  surface  in  the  second 
view,  and  move  the  dividers  along  this  line  until  a  line,  or 
the  projection  of  a  line,  is  found  to  coincide  with  the  other 
point  of  the  dividers.      Examples  of  this  will  appear  later  on. 

In  order  to  aid  the  student  to  read  a  drawing,  we  have 
selected  the  plate  Compound  Rest  and  will  show  in  detail 
by  what  process  of  reasoning  this  drawing  is  read. 

108.  To  find  the  shape  of  the  different  jiarts  and  also 
to  discover,  if  possible,  the  relation  between  them,  we  must 
commence  our  investigation  somewhere.      Let  us  choose  the 

.1/.  E.     v.— II 


68  MECHANICAL  DRAWING  §  U 

bottom  of  the  front  view.  Looking  at  this  it  is  noticed 
that  a  partial  section  is  shown,  from  which,  by  reason  of  the 
section  lining  running  in  opposite  directions,  we  conclude 
that  A  and  B  are  separate  parts.  At  the  right  and  left  of 
the  front  view,  the  full  lines  c,  c  show  that  some  part  of  A 
is  higher  than  the  bottom  of  B,  but  we  do  not  know  whether 
these  lines  denote  the  top  surfaces  of  projecting  parts 
between  which  B  is  fitted,  or  if  c  is  the  top  surface  of  a 
raised  strip  of  some  kind  that  extends  clear  through  the 
inside  of  B.  In  order  to  settle  this  question,  we  note 
whether  the  top  surface  is  continued  somewhere.  Looking 
at  the  front  view  it  is  seen  that  the  line  c  is  dotted  clear 
through  B,  which  settles  conclusively  that  the  part  whose 
top  surface  is  shown  by  the  line  r  is  a  raised  strip  extending 
clear  through  B\  this  fact  immediately  implies  that  B  has  a 
groove  of  some  kind  running  through  it  longitudinally  in 
order  to  admit  the  raised  strip. 

Referring  now  to  the  sectional  view,  which,  as  previously 
stated,  is  a  view  taken  on  the  line  a  b  of  the  front  view,  and 
everything  to  the  right  of  this  line  being  removed,  we  may 
choose  the  bottom  line  d'  of  the  sectional  view  as  a  base 
from  which  to  make  measurements.  From  the  fact  that  the 
section  is  taken  on  the  line  a  b,  we  know  that  the  line  just 
chosen  is  the  projection  of  the  intersection  d  of  the  plane 
represented  hy  a  b  with  the  bottom  of  A.  Measuring  from 
d'  upwards  to  the  highest  line  c'  of  A  in  the  sectional  view, 
and  placing  one  point  of  the  dividers  on  d  in  the  front  view, 
it  will  be  seen  that  the  other  point  coincides  with  the  dotted 
line  forming  a  continuation  oi  c  c\  this  shows  that  c'  is  the 
projection  of  c.  In  a  similar  manner,  we  determine  that  e' 
is  the  projection  of  r,  and  tracing  the  outlines  of  A  in  the 
sectional  view,  we  notice  that  the  raised  strip  on  A  has 
inclined  sides.  "We  also  notice  that  B  is  cut  out  tO'Suit  the 
profile  of  A,  except  that  on  one  side  a  steel  part  L  is  inter- 
posed betAveen  the  inclined  sides  of  A  and  B;  it  is  also  seen 
that  a  screw  rests  with  its  point  against  L. 

Referring  now  to  the  front  view,  and  knowing  from 
inspection  of  the  sectional  view  that   the  upper  and  lower 


§  U  MECHANICAL  DRAWING  69 

surfaces  of  the  steel  part  L  are  flush  with  .'  and  .'   or  c  and  e 
in   the   front   view,    to  determine   the  length   of  this   part 
notice   ,f  any   dotted  or  full   lines    showing   its   length    are 
shown  anywhere   at  a  right  angle  to  c  and  ..      Non;  being 
found    the   conclusion  to  be  drawn  is  that  either  the  steel 
part  /.  IS  as  long  as  A  or  has  the  same  length  as  B     A  per 
son  without  any  practical  experience   migi^t   conclude  that 
the  length   is  the  same  as  that  of  J;  but  any  one  havino- 
engineenng    instinct  or  practical    knowledge   would  imme^ 
diately  notice   that,  as  the  steel  strip  has  setscrews  which 
evident  y  serve  to  push  it  against  the  inclined  side  of  A    it 
would   be  unnecessary  to  make  the  strip  the  leno-th  of '-J 
and  hence  would  immediately  conclude  that  its  length  is  the 
same  as  that  of  />'.      This  latter  conclusion  is  th?  one  the 
arattsman  desired  to  convey. 

109.     Looking  at    the    sectional   view   of    A    again    we 
notice  that  a  groove,  open  on  top,  is  cut  into  A.     To  find  its 
length  we  must  find  lines  corresponding  to  it  in  the  front 
view.      Measuring  from  d'  upwards  to  the  bottom  of  the 
groove  and  transferring  the  measurement  to  the  front  view 
we  find  that  the  dotted  line,.,^  represents  the  bottom  and 
end  of    the  groove,  which  at  the  left  is  also  shown  to  be 
open  at  the  bottom,  since   the  dotted  line  ,,^  curves  around 
and  continues  to  the  bottom  of  A.     This  is  also  indicated 
by  the  dotted  hues  <^'  that  form  an  extension  of  the  sides  of 
the  groove  in  the  sectional  view;  measuring  from  ^' down- 
wards to  the  horizontal  dotted  line  joining  the  ends  of  <^'  <r' 
and  then  passing  along  .  in  the  front  view,  the  point  of  the 
dividers  will  be  found  to  coincide  with  the  point  where  the 
dotted  hue   .r  „,eets  the  bottom  of  A.      From   this  the  con- 
clusion IS  drawn  that  the  dotted  horizontal  line  joining  ^r'  .' 
ni  the  sectional  view  is  the  bottom  edge  of  the  opening!    " 

no      Looking  at  the  front  view  we  notice  a  left-handed 
screws  that   is  placed   within   the   slot   just  investigated 
Knowing  that,  in  a  view  m  line  with  its  axis,  the  outline  of 
a  screw  will  be  a  circle,  and  knowing  that  this  circle  will  be 
found   inside  of  the  slot,  the  screw  is  readily  found  in  the 


70  MECHANICAL  DRAWING  §  U 

sectional  view.  Xow,  experience  teaches  us  that  when  a 
screw  is  shown  in  place  in  a  machine  drawing,  there  must 
also  be  somewhere  along  its  axis  a  threaded  hole  (or  a  nut) 
to  receive  it.  Looking  at  the  sectional  view  we  see  the  out- 
line of  something  (marked  '•'■Bronze")  that  surrounds  the 
screw.  Now,  this  part,  at  first  glance,  appears  to  be  a  con- 
tinuation of  the  pin  E  directly  above  it;  there  are  two 
reasons,  however,  why  this  is  not  the  case.  In  the  first 
place,  the  part  E  is  sectioned  for  steel;  this  immediately 
shows  that  E  and  the  part  under  investigation  are  separate 
parts.  Furthermore,  when  tracing  out  the  shape  and  posi- 
tions of  the  objects  in  the  front  view,  they  will  be  seen 
neither  to  be  in  line  nor  to  have  any  connection  with  each 
other.  To  find  the  part  under  investigation  in  the  front 
view,  we  may  take,  in  the  sectional  view,  a  measurement 
from  the  center  of  the  screw  downwards  to  the  lowest  point 
of  the  part  we  are  investigating,  and  then,  referring  to  the 
front  view,  proceed  along  the  center  line  of  the  screw  C 
until  we  strike  the  dotted  line  //.  Since  the  sectional  view 
shows  only  things  to  the  left  of  the  line  a  h,  we  know  that 
as  the  part  being  investigated  shows  in  the  sectional  view, 
we  must  look  for  it  to  the  left  oi  a  b  in  the  front  view.  At 
the  ends  of  the  horizontal  dotted  line  //  we  notice  two  verti- 
cal dotted  lines  that  show  the  length  of  the  part  under  inves- 
tigation; since  these  lines  terminate  against  the  liner,  we 
know  that  the  part  butts  against  the  surface  of  B. 

This  latter  conclusion  is  further  confirmed  by  examining, 
in  the  sectional  view,  the  full  outline  of  the  part.  Referring 
again  to  the  front  view,  we  see  a  screw  thread  indicated  in  B 
right  above  the  part  we  are  discussing,  and  in  the  absence 
of  any  indication  to  the  contrary  may  justly  assume  that  it 
is  a  threaded  shank  by  means  of  which  the  part  is  attached 
to  B.  By  this  time  we  are  probably'  convinced  that  the 
part  we  have  been  investigating  is  the  nut  we  are  looking 
for,  but  are  not  sure  of  it.  To  find  out,  let  us  try  to  inves- 
tigate the  whole  of  the  screw.  In  the  front  view,  the  dot- 
ted lines  i,  i  show  that  the  screw  has  a  bearing  and  also  has 
a  collar  butting  against  part  of  A  ;  beyond  the  bearing  the 


§  U  MECHANICAL  DRAWING  71 

screw  shows  in  full  and  apparently  has  a  seat  for  some 
kind  of  an  attachment  which  must  cause  the  screw  to  turn, 
since  a  dowel-pin  is  shown  in  the  seat.  Inspecting  the  sec- 
tional view  we  find  a  screw  similar  to  the  one  under  dis- 
cussion, with  a  ball  handle  and  retaining  nut  on  the  end  of 
it.  As  we  find  a  note  "  l\vo  Handles  Machinery  Steel, 
Finish  all  over;'  and  as  we  cannot  find  any  other  place  for 
the  second  handle,  we  naturally  conclude  that  such  a  handle 
is  to  be  placed  on  the  end  of  the  screw  C.  Now,  from  the 
fact  that  the  screw  is  confined  longitudinally  by  the  collar 
and  the  ball  handle,  and  that  there  is  no  thread  on  the 
part  of  the  screw  between  them,  we  know  that  the  nut 
must  be  ta  the  right  of  the  collar;  since  the  part  previously 
investigated  is  the  only  part  we  can  find  that  directly  sur- 
rounds the  screw,  we  will  now  be  justified  in  assuming  that 
it  is  the  nut  we  are  looking  for. 

111.  The  fact  that  A  and  B  are  connected  together  by 
a  screw  provided  with  a  handle  for  turning  it  will  immedi- 
ately suggest  the  idea  that  it  is  to  be  used  for  moving  the 
part  B  along  A,  whence  we  conclude  that  B  is  a  slide 
moving  on  A.  Knowing  this,  the  logical  conclusion  is  that 
the  piece  Z  is  a  gib  used  for  taking  up  the  wear  of  the  sliding 
surfaces,  Avhich  view  is  proved  to  be  correct  when  it  is 
noticed,  by  reference  to  the  sectional  view,  that  a  tighten- 
ing of  the  setscrew  will  tend  to  draw  the  wearing  surfaces 
together. 

112.  Looking  now  at  the  part  K,  at  the  left  of  A,  in  the 
front  view,  considering  the  part  by  itself,  we  cannot  tell 
whether  it  is  an  integral  part  of  A  or  a  separate  piece  fa.st- 
ened  to  it.  But  as  soon  as  we  consider  it  in  connection 
with  the  screw  (T,  we  see  that  the  latter  cannot  be  placed  in 
position  unless  the  part  K  is  removable.  From  this  we 
conclude  that  the  part  K  is  separate  from  A.  The  next 
question  that  suggests  itself  is:  How  is  it  fastened  on? 
The  note  on  the  front  view  and  the  dotted  screw  heads  in 
the  sectional  view  show  that  screws  with  slotted  heads  are 
used. 


72  MECHANICAL  DRAWING  |  U 

113.  As  far  as  the  shape  of  A'  is  concerned,  the  front 
view  shows  that  it  is  a  cone  joining  some  presumably  flat 
part.  Referring  now  to  the  sectional  view,  we  discover  by 
measuring  successively  from  the  center  line  and  center  of 
the  screw  C  that  the  dotted  horizontal  line  k'  represents 
the  lower  surface  of  K,  and  the  absence  of  any  other  dotted 
lines  in  this  part  of  the  sectional  view  indicates  that  the 
profile  of  the  flat  part  of  A'  is  the  same  as  that  of  A. 

114.  On  examining  the  sectional  view,  it  is  seen  that 
some  part  of  D,  which  from  the  section  lining  we  know  to 
be  separate  from  B,  projects  downwards  from  the  main  body 
of  D  and  is  in  contact  with  the  upper  surface  /'  of  the 
part  B.  Referring  now  to  the  front  view  and  looking 
along  /.  we  find  that  the  part  under  discussion  is  cylindrical ; 
this  is  inferred  Trora  the  dimension  "4'  turned."  The  main 
body  of  D,  and  also  the  parts  G,  H.  and  J,  may  now  be 
investigated  in  a  manner  similar  to  that  in  which  the 
relation  of  A,  B,  C,  and  AT  was  traced;  it  will  then  be 
found  that  A>  is  a  part  similar  to  A.  Furthermore,  the 
investigation  will  show  that  6^  is  a  slide;  this  slide  is 
movable  by  means  of  the  screw  //.  which  turns  in  the 
bearing  y. 

115.  Referring  again  to  the  sectional  view,  we  see  that 
B  and  D  are  connected  together  by  a  pin  £,  whose  purpose 
is  unknown  as  yet.  Examining  this  pin  we  notice  that  a 
hole  is  cut  through  its  upper  end  and  that  a  screw  F,  with 
a  tapered  shoulder  to  the  right  of  its  screw  thread,  passes 
through  this  hole.  On  close  examination,  we  see  that  the 
hole  in  £  is  so  placed  that  the  tapered  part  of  the  screw  F 
bears  against  the  upper  side  of  the  hole.  We  further  notice 
that  the  screw  F  is  not  used  as  a  fastening  device  to  hold 
any  parts  of  D  together ;  this  conclusion  is  forced  upon  us 
by  the  fact  that  the  sectional  view  shows  D  to  be  one  piece. 
Now,  we  know  from  experience  that  a  screw  is  used  either 
as  a  fastening  device  or  to  transmit  motion ;  as  it  obviously 
is  not  used  for  the  purpose  first  mentioned,  we  conclude 
that  it  probably  serves  for  the  latter  purpose.     To  make 


§  n  MECHANICAL  DRAWING  73 

sure  of  this  we  trace  out  what  will  happen  if  the  screw  is 
rotated.  We  then  noticed  that  if  the  screw  is  screwed 
inwards,  it  will  raise  the  part  A";  but  as  £  cannot  move 
upwards  by  reason  of  being-  confined  by  the  collar  on  it,  it 
shows  to  us  that  screwing  F  inwards  will  force  D  down 
on  B.  The  logical  inference  is  that  /i  and  F  iorm  a  clamp- 
ing device  intended  to  clamp  B  and  D  together. 

Examining  the  pin  F  again,  we  do  not  find  anything  that 
would  definitely  tell  whether  it  is  round  or  square.  Here 
judgment  must  be  used.  An  experienced  person  would 
know  upon  the  first  glance  that  the  clamping  arrangement 
shown  is  an  expensive  one  to  make  and  one  not  likely  to  be 
adopted  when  it  is  only  required  to  fasten  two  pieces 
rigidly  together,  in  which  case  F  might  be  either  round  or 
square.  The  next  inference  would  be  that  it  is  used  in 
order  to  allow  D  to  be  rotated  around  F  and  to  be  clamped 
in  any  position.  This  supposition  requires  the  pin  F  to  be 
round  and  is  correct  in  this  case. 

116.  Referring  now  to  the  ball  handle  /,  of  which  only 
one  view  is  shown,  the  question  of  whether  it  is  circular  or 
square  is  immediately  settled  by  experience  teaching  us  that 
a  handle  having  the  shape  shown  is  not  likely  to  be  any- 
thing else  but  round,  and  in  the  absence  of  any  note  or  indi- 
cation to  the  contrary,  we  would  be  justified  in  assuming  it 
to  be  round. 

117.  As  far  as  the  part  G  is  concerned,  the  sectional 
view  shows  it  to  be  cored  out  in  order  to  pass  over  the  nut 
in  which  the  screw  H  Avorks.  The  width  and  profile  of  the 
coring  must  be  obtained  from  the  front  view,  which  it  will 
be  remembered  is  a  view  at  a  right  angle  to  the  sectional 
view.  The  natural  assumption  to  make  is  that  the  lines 
giving  the  width  and  profile  of  the  coring  will  be  found 
directly  in  the  vicinity  of  the  screw  //  in  the  front  view. 
Measuring  from  the  center  line  of  this  screw  in  the  sec- 
tional view  upwards  to  the  line  showing  the  height  of  the 
coring,  and  then  transferring  this  measurement  to  the  front 
view,  we  find  the  full  circle  //.    Now,  as  the  coring  is  beyond 


74  MECHANICAL  DRAWING  §  14 

the  bearing  J,  we  know  that  its  profile  would  show  in  dotted 
lines  and  conclude  that  the  circle  //  represents  some  part  of 
the  bearing  J.  As  this  bearing  has  a  conical  projection, 
the  inference  is  that  the  full  circle  represents  the  largest 
diameter  of  the  cone,  which  is  the  case.  Now,  the  absence 
of  a  dotted  line  showing  the  coring  forces  us  to  conclude  that 
the  dotted  line  would  be  directly  behind  the  full  circle  ;/  and 
is  thus  hidden.  This  conclusion  is  further  strengthened  by 
finding  two  vertical  dotted  lines  r,  r  tangent  to  the  circle  7/, 
and  we  finally  decide  that  the  groove  has  straight  sides  with 
a  semicircular  top,  as  given  by  the  dotted  lines  r,  r  and  the 
upper  semicircle  of  n.  By  measuring  again  in  the  manner 
previously  explained,  we  decide  that  the  dotted  line  t?  is  a 
front  view  of  the  nut  in  which  H  works. 

118.  At  the  right-hand  end  of  the  sectional  view  of  G 
we  notice  a  T-shaped  opening.  Referring  to  the  front  view 
we  can  easily  discover,  hy  transferring  measurements,  that 
the  dotted  horizontal  lines  /,  /  show  the  length  of  the  slot, 
which  is  seen  to  extend  clear  across  G. 

119.  Referring  now  to  the  drawing  of  the  tool  post,  it 
will  be  observed  that  only  one  view  is,  given.  While  this 
does  not  definitely  settle  that  the  post  is  circular  in  cross- 
section,  common  practice  would  justify  a  person  in  assu- 
ming, in  the  absence  of  any  note  or  any  other  indication  to 
the  contrary,  that  such  was  the  case.  This  view  is  strength- 
ened by  the  fact  that  some  dimensions  are  marked  d,  signify- 
ing diameter,  which  term  is  rarely  applied  to  any  but  a 
round  object. 

1*20.  The  two  v^ews  of  the  collar  give  its  shape.  Refer- 
ring to  the  front  view,  while  there  is  no  definite  note  to  that 
effect,  it  would  be  inferred  from  the  fact  that  a  thread  is 
shown  that  the  lower  part  is  separate,  being,  in  fact,  a 
circular  nurled  nut  threaded  to  receive  the  upper  part. 

121.  While,  generally  speaking,  any  one  can  learn  to 
determine  the  shape  of  objects  from  a  drawing,  there  are 
cases   that   arise   in   practice    where   this    is    very   difficult 


•-sTTcsziz-c-n  cr'Scyezvs. see 


fa's 


COMPOUND   REST 
Fdr  idx 36  "Speed  Lathe. 

Cast  IrtON,  Unless  Orj-iEiP.wfSE  SpEcineo, 

OneOfThis-  Scale Fuu.  Size. 

The  3cr ANTON  Tool  Works, 

ScRAf^TCM,  Pa. 
Or-Offn  By  <JA  G-  D ra^n'SJ'  182  ■ 
Checked  3</CPT.  Order /A  9r. 

D^TTE   0CT/-/900. 


-^/5 

•6c  a^ss-z:-:. 

-^^ 

IWTE  85.189  3 


G 


!l  J 


\ 

'^    Jfarden  end,  of^creio. 


/6  Threads 


/  % 


U\\ 


.F 


Rf 


-//^ 


T^'t^""^" 


iffg 


:^^=^ 


<**> 


■ned 


T^i — -^j ^ 

-*-* — '— "^ r-i — y,« 


,^/Z^.^,. 


-l^^•ll'i., 


TOOL  POST. 
One  a/d?tT.5,Tool  Steel. 


% 


/O 


f----X:il^jiyFl£Cce  o?2  cen  ter  lz/?zg . 


^76 


4^ 


Ta,;o 


-//7s 


-tofiitdecZ'- 


0?z^oft^ts.  ToalSieel, 


fH^ 


III     1  1 


"t?V^  -r. 


%-.--"mm 


N'/S63 


ediately  following  the  title  page. 


J-0^f7\rjMITH.CZA;5SJi945Z9. 


§  14  MECHANICAL  DRAWING  75 

without  further  verbal  or  written  instructions.  The  cases  in 
which  this  usually  happens  are  where  coring  has  various  odd- 
shaped  curved  surfaces  that  curve  in  different  directions,  as 
occurs,  for  instance,  with  the  steam  ports  and  other  pas- 
sages of  steam-engine  cylinders  and  other  similar  work. 
Practical  experience  with  a  certain  line  of  work,  and, 
frequently,  a  knowledge  of  the  object  of  the  doubtful 
part,  will  often  allow  the  reader  to  form  a  correct  idea  of 
Avhat  the  draftsman  is  trying  to  convey;  when  this  experi- 
ence or  knowledge  is  lacking,  consult  somebody  who  is  likely 
to  know. 

Furthermore,  the  shape  of  an  object  does  not  necessarily 
in  itself  always  reveal  its  purpose.  Ability  to  determine  at 
sight  what  an  object  is  to  be  used  for  involves  either  a 
thorough  knowledge  of  a  particular  line  of  work — in  which 
case  the  purpose  of  objects  coming  within  its  range  can 
usually  be  determined  at  sight — or  a  very  wide  general 
knowledge  of  engineering  construction. 


DRAWING    PLATE,  TITLE  :     COMPOUJ^D    REST 

13!3.  This  drawing  is  at  one  and  the  same  time  a  detail 
and  an  assembly  drawing,  as  it  not  only  gives  all  dimensions 
required  for  each  and  every  part,  but  also  shows  how  they 
are  assembled.  Drawings  are  frequently  made  in  this 
manner  in  order  to  save  some  of  the  draftsman's  time.  The 
particular  drawing  shown  exhibits  a  case  where  the  draftsv 
man,  on  account  of  lack  of  space,  has  been  compelled  to 
break  the  rules  governing  the  arrangements  of  the  views. 
Referring  to  the  plate,  the  sectional  view  taken  on  the  line  a  b 
should  have  been  placed  alongside  and  on  the  right  of  the 
front  view,  which  obviously  cannot  be  done  without  making 
the  drawing  to  a  scale  smaller  than  full  size. 

1^3.  To  begin  locate  the  center  line  of  the  screw  C  in 
the  front  view  at  a  distance  of  4^  inches  from  the  iq^per 
border  line.  The  slide  B  being  movable  along  the  base  A^ 
it  can  be  located  anywhere  within  its  range  of  movement; 


76  MECHANICAL  DRAWING  §  U 

but  in  order  to  get  a  convenient  arrangement,  it  is  recom- 
mended to  locate  the  line  a  b  passing  through  the  center  of 
the  screw  of  the  upper  slide  at  a  distance  of  '\\\  inches  from 
the  left-hand  border  line.  Locate  the  center  line  of  the  hole 
in  the  base  marked  |"  tap  lly\  inches  from  the  left-hand 
border  line.  The  front  view  can  now  be  drawn  from  the 
dimensions  given. 

12-4,  Locate  the  center  line  of  the  screw  H  in  the  sec- 
tional view  8^  inches  from  the  upper  border  line  and  locate 
the  center  line  of  the  handle  /at  a  distance  of  "^f  inches  from 
the  left-hand  border  line.  The  upper  slide  G  being  movable 
on  its  base  D,  the  latter  could  be  drawn  anywhere  within 
the  range  of  movement ;  in  order  to  have  a  good  arrange- 
ment, however,  it  is  recommended  to  locate  the  vertical 
center  line  passing  through  the  part  ^  at  a  distance  of 
VlW  inches  from  the  left-hand  border  line.  The  sectional 
view  can  now  be  drawn  without  any  special  instructions. 

125.  The  center  line  of  the  collar  should  be  located  at  a 
distance  of  If  inches  from  the  right-hand  border  line  and  the 
center  of  the  top  view  of  the  collar  is  to  be  located  at  a  dis- 
tance of  4r|  inches  from  the  upper  border  line. 

126.  The  center  line  of  the  tool  post  may  be  located  at 
a  distance  of  Iff  inches  from  the  upper  border  line  and  the 
right-hand  end  of  the  tool  post  at  a  distance  of  If  inches 
from  the  right-hand  border  line.  The  tool-post  screw  being 
movable,  it  could  be  shown  in  any  position  within  its  range 
of  movement ;  but  in  order  to  get  the  same  arrangement  as 
on  the  plate,  the  right-hand  surface  of  its  collar  should  be 
placed  at  a  distance  of  ^  inch  from  the  left-hand  end  of  the 
body  of  the  tool  post. 

A  number  of  reference  letters  printed  in  bold-face  italics 
appear  on  this  plate.  The  student  should  omit  these  letters 
on  his  drawing. 

127.  In  order  to  emphasize  the  fact  that  different  drafts- 
men prefer  difi"erent  styles  of  lettering  and  a  different 
arrangement  for  titles,  this  plate  has  been  given  a  title  dif- 
fering from  those  previously   given.      The  first   and  second 


^  U  MECHANICAL  DRAWING 


77 


lines  of  lettering  are  easily  made  with  tlie  ruling  pen,  the 
inclination  of  the  letters  being  75^  The  letters  in  the  'first 
line  of  lettering  are  ^\  inch  high;  in  the  second  line  all 
capitals  and  numerals  are  -L  inch  high,  except  the  llrst  letter 
of  each  word,  which  is  -/V  inch  high.  Tin-  third  to  ninth  line 
of  lettering  are  written  freehand.  In  the  third  and  fourth 
lines,  the  lettering  is  ^\  inch  high,  except  the  first  letter  of 
each  word,  which  is  ^  inch  high.  In  the  fifth  line,  all  letters 
are  i  inch  high,  except  that  the  first  letter  of  each  word  is 
^%  inch  high.  The  sixth  line  is  the  same  as  the  third  and 
fourth.  In  the  remaining  lines,  the  capitals  are  ^\  inch  and 
the  small  letters  -^\  inch  high.  The  heading  "Tool  Post  " 
is  ^  inch  high. 

TRACINGS 

128.  In  actual  practice  in  the  drawing  room,  it  is 
necessary  to  have  more  than  one  copy  of  a  drawing.  It 
would  be  very  expensive  to  make  a  finished  drawing  every 
time  an  extra  copy  was  wanted  and  to  avoid  this  tracings 
and  blueprints  are  made.  Any  number  of  blueprint  copies 
can  be  made  from  the  same  tracing.  A  complete  pencil 
drawing  is  made  first ;  then,  instead  of  inking  in  as  hereto- 
fore, a  piece  of  tracing  paper  or  tracing  cloth  of  the  same 
size  as  the  pencil  drawing  is  fastened  to  the  board  over  the 
original  drawing.  The  tracing  paper  or  cloth  being  almost 
transparent,  the  lines  of  the  drawing  can  be  readily  seen 
through  it,  and  the  drawing  is  inked  in  on  the  tracing  paper 
or  cloth  in  the  same  manner  as  if  inking  in  a  finished 
drawing. 

129.  Ti-aeing  paper  is  but  little  used  in  this  country. 
It  is  easily  .torn  and  cannot  be  preserved  as  well  as  tracing 
cloth.  The  two  sides  of  the  tmciiis  cloth  are  known  as 
the  glazed  side  and  the  dull  side;  they  are  also  known  as 
the  front  and  the  back.  The  glazed  side,  or  front,  is  covered 
with  a  preparation  that  gives  it  a  very  smooth  polished 
surface;  the  back,  or  dull  side,  has  very  much  the  appear- 
ance of  a  piece  of  ordinary  linen  cloth.      Either  side  may  be 


78  MECHANICAL  DRAWING  §  U 

used  for  drawing  upon,  but  when  the  glazed  side  is  used, 
care  must  be  taken  to  remove  all  dirt  and  grease,  otherAvise 
the  ink  will  not  flow  well  from  the  pen.  This  can  be  done 
by  taking  a  knife  or  a  file  and  scraping  or  filing  chalk  upon 
the  tracing  cloth;  then  take  a  soft  rag  of  some  kind — cotton 
flannel  or  chamois  skin — and  rub  it  all  over  the  tracing 
cloth,  being  sure  to  rub  chalk  over  ever}^  spot.  Finally,  dust 
the  rag  and  remove  as  much  of  the  chalk  from  the  cloth  as 
can  be  gotten  off  by  rubbing  with  the  rag.  The  finer  the 
chalk  powder  is  the  better.  It  is  not  usual  to  chalk  the 
dull  side,  but  it  improves  it  to  do  so.  The  glazed  side  takes 
ink  much  better  than  the  dull  side,  the  finished  drawing  looks 
better  and  will  not  soil  so  easily,  and  it  is  also  easier  to  erase 
a  line  that  has  been  drawn  on  this  side.  Pencil  lines  can  be 
more  satisfactorily  drawn  on  the  dull  side,  and  if  it  is  desired 
to  photograph  the  drawing,  it  is  better  to  draw  on  this  side. 
The  draftsman  uses  either  side,  according  to  the  work  he  is 
doing  and  to  suit  his  individual  taste,  but  if  the  glazed 
side  is  used,  /'/  must  be  cJialkcd.  The  tracings  are  drawn 
in  a  manner  similar  to  the  finished  drawings,  the  center 
lines,  section  lines,  etc.  being  drawn  exactly  as  previously 
described.  In  some  oflfices  it  is  customary  to  draw  the 
center  lines  and  dimension  lines  on  a  tracing  in  red  ink,  so 
that  they  may  appear  gray  instead  of  white  on  the  blueprint. 

130.  After  having  drawn  the  plate  entitled  Compound 
Rest,  the  student  is  required  to  make  a  tracing  of  the  plate 
entitled  Turret-Lathe  Tools. 


BLirEPRIXTIKG 

131.  Blueprinting  is  the  process  of  duplicating  a  tracing 
by  means  of  the  action  of  light  upon  a  sensitized  paper. 
The  following  solution  is  much  used  for  sensitizing  the 
paper  :  Dissolve  2  ounces  of  citrate  of  iron  and  ammonia  in 
8  ounces  of  water ;  also  1^  ounces  of  red  prussiate  of  potash 
in  8  ounces  of  water.  Keep  the  solutions  separate  and  in 
dark-colored   bottles  in  a  dark  place  where  the  light  cannot 


§  li  MECHANICAL  DRAWING  79 

reach  them.      Better  results   will  be  obtained  if  ^  an  ounce 
of  gum  arabic  is  dissolved  in  each  solution. 

When   ready  to   prepare  the  paper,  mix  equal  portions  of 
the  two  solutions,  and  be  particularly  careful  not  to  allow 
any    more  light    to  strike   the   mixture   than  is   absolutely 
necessary    to    see  by.      For    thi^  reason,  it  is  .necessary  to 
have  a  dark  room  to  work  in.      There  must  be  in  this  room 
a  tray  or  sink  of  some  kind  that  will   hold  water;  it  should 
be    larger    than   the    blueprint    and    about    6    inches   deep. 
There  should  also  be  a  flat  board  large  enough  to  cover  the 
tray  or  sink.     If  the  sink   is  lined  with   zinc  or  galvanized 
iron,  so  much  the  better.      There   must  be  an  arrangement 
like  a  towel  rack  to  hang  the  prints  on  while  they  are  drying. 
For  the  want  of  a   better  name,  this  arrangement   will  be 
called  a  print  rack.     The  paper  used  for  blueprinting  should 
be  a  good,  smooth,  white   paper,  and   may  be   purchased  of 
any  dealer  in  drawing  materials.     Cut  it  into  sheets  a  little 
larger  than  the   tracing,  so  as  to  leave  an  edge  around  it 
when  the  tracing  is  placed  upon  it.     Place   eight  or  ten  of 
these  sheets  upon  the  fiat   board  before  mentioned,  taking 
care  to  spread  flatly  one  above  another,  so  that  the  edges  do 
not  overlap.     Secure  the  sheets  to  the  board  by  driving  a 
brad   or   small    wire    nail    through  the  two    upper   corners 
sufficiently  far    into  the  board    to   hold  the   weight  of  the 
papers  when  the  board  is  placed  in  a  vertical  position.     Lay 
the  board  on  the  edges  of  the  sink,   so  that    one    edge  is 
against  the  wall  and  the  board  is  inclined  so  as  to  make  an 
angle  of  about  00°  with  the   horizontal.      Darken  the  room 
as  much  as  possible  and  obtain  what  light  may  be  necessary 
from  a   lamp  or  gas  jet,  which  should  be  turned  down  very 
low.     With  a   wide    camel's-hair   brush    or   a    fine    sponge, 
spread  the  solution  just  prepared  over  the  top  sheet  of  paper. 
Be  sure  to  cover  every  spot  and  do  not  get   too  much  on 
the    paper.     Distribute    it    as  evenly   as    possible    over  the 
paper,  in  much  the  same  manner  that  the   finishing  coat  of 
varnish  would  be  put  on   by  a  painter.      Remove   the  sheet' 
by  pulling  on  the  lower  edge,  tearing  it  from  the  nail  that 
holds  it,  and  place  it  in  a  drawer  where  it  can  lie  fiat  and  be 


80 


MECHANICAL  DRAWING 


§1^ 


kept  from  the  light.  Treat  the  next  sheet  and  each  succeed- 
ing sheet  in  exactly  the  same  manner,  until  the  required 
number  of  sheets  has  been  prepared. 

Unless  a  large  number  of  prints  is  constantly  used,  it  is 
cheaper  to  buy  the  paper  already  prepared.  It  can  be  bought 
in  rolls  of  10 -yards  or  more,  of  any  width,  or  in  sheets  already 
cut  and  ready  for  use.  There  is  very  little,  if  anything, 
saved  in  preparing  the  paper,  and  better  results  are  usually 
obtained  from  the  commercial  sensitized  paper,  since  the 
manufacturers  have  machines  for  applying  the  solution  and 
are  able  to  distribute  it  very  evenly. 

132.  In  Figs.  15  and  16  are  shown  two  views  of  a  print- 
ing frame  that  is  well  adapted  to  sheets  not  over  17'  X  "21". 
The  frame  is  placed   face   downwards    and  the   back   A  is 


Fig.  15 


removed  by  unhooking  the  brass  spring  clips  ^,  B  and  lifting 
it  out.  The  tracing  is  laid  upon  the  glass  C,  with  the  inked 
side  touching  the  glass.  A  sheet  of  the  prepared  paper,  per- 
fectly dry,  is  laid  upon  the  tracing  Avith  the  yellow  (sensitized) 


§  1-t 


MECHANICAL  DRAWING 


81 


side  downwcirds.  The  j)aper  and  tracing  are  smoothed 
out  so  as  to  lie  perfectly  flat  upon  the  glass,  the  cover  A  is 
replaced,  and  the  brass  spring  clips  B,  B  are  sprung  under 
the  plates  D,  so  that  the  back  cannot  fall  out.  While  all 
this  is  being  done,  the  paper  should  be  kept  from  the  light 
as  much  as  possible.  The  frame  is  now  placed  where  the 
sun  can  shine  upon  it  and  is  adjusted,  as  shown  in  Fig.  16,  so 
that  the  sun's  rays  will  fall  upon  it  as  nearly  at  right  angles  as 
possible      According  to  the  conditions  of  the  sky — whether 


Fig.  16 
clear  or  cloudy — and  the  time  of  the  year,  the  print  must 
be  exposed  from  3  to  15  minutes.  The  tray,  or  sink,  already 
mentioned,  shoukl  be  filled  to  a  depth  of  about  'I  inches  with 
clear  water  (rain  water  if  possible).  The  print  having  been 
exposed  the  proper  length  of  time,  the  frame  is  carried  into 
a  dark  part  of  the  room,  the  cover  removed,  and  the  print 
(prepared  paper)  taken  out.  Now  place  it  on  the  water 
with  the  yellow  side  down  and  be  sure  that  the  water 
touches  every  part  of  it.  Let  it  soak  while  putting  the  next 
print  in  the  frame.     Be  sure  that  the  hands  are  dry  before 


82  MECHANICAL  DRAWING  §  U 

touching  the  next  print.  The  first  print  having  soaked  a 
short  time  (about  10  minutes)  take  hold  of  two  of  its 
opposite  corners  and  lift  it  slowly  out  of  the  water.  Dip  it 
back  again  and  pull  out  as  before.  Repeat  this  a  number  of 
times,  until  the  paper  appears  to  get  no  bluer;  then  hang  it 
by  two  of  its  corners  to  dry  on  the  print  rack  previously 
mentioned.  If  there  are  any  dark-purple  or  bronze-colored 
spots  on  the  prints,  it  indicates  that  the  prints  were  not 
washed  thoroughly  on  those  spots.  If  these  spots  are  well 
washed  before  the  print  is  dried,  they  will  disappear. 

133.  It  is  best  to  judge  the  proper  time  of  exposure 
to  the  light  by  the  color  of  the  strip  of  print  projecting 
beyond  the  edge  of  the  tracing.  To  obtain  the  exact  shade 
of  the  projecting  edge,  take  a  strip  of  paper  about  12  or 
14  inches  long  and  3  or  4  inches  wide.  Divide  it  into,  say, 
12  equal  parts  by  lead-pencil  marks,  and  with  the  lead 
pencil  number  each  part  1,  2,  3,  etc.  Sensitize  this  side  of 
the  paper  and,  after  it  has  been  properly  dried,  place  it  in 
the  print  frame  with  the  sensitized  side  and  the  marks  and 
figures  against  the  glass.  Expose  the  whole  strip  to  the 
light  for  one  minute;  then  cover  the  part  of  the  strip 
n^arked  1  with  a  thin  board  or  anything  that  will  prevent 
the  light  from  striking  the  part  covered.  At  the  end  of  the 
second  minute,  cover  parts  2  and  1 ;  at  the  end  of  the  third 
minute,  parts  3,  2,  and  1,  etc.  When  twelve  minutes  are 
up,  part  1  will  have  been  exposed  one  minute;  part  2,  two 
minutes,  etc.,  part  12  having  been  exposed  twelve  minutes. 
Remove  the  frame  to  a  dark  part  of  the  room  and  tear  the 
strip  so  as  to  divide  it  into  two  strips  of  the  same  length 
and  about  half  the  original  width.  Wash  one  of  the  strips 
as  before  described,  and  Avhen  it  has  dried,  select  a  good 
rich  shade  of  blue,  neither  too  light  nor  too  dark;  notice  the 
number  of  the  part  chosen,  and  it  will  indicate  the  length  of 
time  that  the  print  >vas  exposed.  Examine  carefully  the 
corresponding  part  of  the  other  strip,  and  the  correct  color 
of  the  edge  of  the  print  projecting  beyond  the  tracing  is 
determined.     All  prints  should  be  exposed  until  this  color  is 


§  14 


MECHANICAL  DRAWING 


83 


reached,  no  matter  how  long  or  how  short  the  time  may  be; 
then  they  should  be  immediately  taken  out  and  washed. 


Fig.  17 


134.     In  Fig.  17  is  shown  a  patented  frame  which  can  be 
shoved  out  of  the  window  and  adjusted  to  any  angle.     Wh 
M.  E.     y.—i3 


en 


84  MECHANICAL  DRAWING  §  U 

not  in  use.  it  can  be  folded  up  against  the  wall  and  occupies 
but  little  space.  It  is  made  in  different  sizes  from  16"  X  24' 
to  48' X  TS".  It  is  one  of  the  best  frames  in  the  market, 
and  is  placed  in  such  a  position  relatively  to  the  window  that 
the  window  can  be  lowered  to  the  top  of  the  main  arm.  when 
it  is  desired  to  keep  out  the  cold  during  the  winter. 


DRAWTS'G     PLATE,    TITLE:     SIX-HOKSEPOWER 
HORIZOXTAE    STEA3I    EXGI^'E 

135.  Instead  of  making  a  finished  drawing  of  this 
engine,  as  in  the  previous  plates,  from  an  exact  copy,  the 
student  is  given  the  rough  sketches  of  the  details  of  a  six- 
lioi-sepoTver  liorizontal  steam  engine,  with  full  dimen- 
sions marked  upon  them ;  from  these  he  is  expected  to  make 
a  general  pencil  drawing  of  the  engine  in  two  views — a  plan 
and  a  side  elevation.      The  details  are  not  to  be  drawn. 

The  pencil  drawing  should  then  be  traced  according  to 
the  directions  previously  given.  The  details  are  not  drawn 
to  scale,  but  are  fully  dimensioned.  In  order  to  draw  the 
engine  to  as  large  and  as  convenient  a  scale  as  possible,  it  is 
necessary  to  make  this  tracing  a  trifle  larger  than  the  plates 
that  have  preceded  it.  The  size  over  all  will  be  14+' X  18|', 
with  the  usual  border  line  ^  inch  from  each  edge  all  around. 

That  the  student  may  have  a  good  idea  of  what  he  is 
expected  to  do,  a  greatly  reduced  cut  of  the  general  draw- 
ing is  also  given  him.  All  dotted  lines  indicating  parts  not 
seen  have  been  omitted  in  order  to  simplify  the  work.  The 
scale  to  be  used  is  3  inches  =  1  foot. 

136,  Draw  the  center  lines  inn,  pq,  rs,  and  ti:  Draw 
the  side  elevation  of  the  bedplate  with  the  bearing  caps  in 
position,  from  the  dimensions  given  on  the  detail  sketches, 
taking  care  to  make  the  parts  that  are  likely  to  be  hidden  by 
the  flywheel,  eccentric  rod,  etc.  light  so  that  they  may  be 
easily  erased  before  tracing.  The  drawing  may  be  traced 
without  removing  the  unnecessary  construction  lines,  but  it 
is  better  to  do  so,  since  it  lessens  the  liability  of  inking  in 
lines  that  will  have  to  be  erased  from  the  tracinor. 


l\\f^^^=^ 


JU^E  25. 1893. 


jT  notice  cf  copvT:g''-t,  se 


ediately  following  the  title  page. 


J-OHJvsjyriTj^.  CLA  ss  jsr^  Asag. 


§  14  MECHANICAL  DRAWING  85 

Draw  the  plan  of  the  bedplate  with  the  bearing  caps, 
studs  and  nuts,  foundation-bolt  holes,  etc.  shown  in  their 
proper  places  and  positions.  The  different  curves  of  the 
bearing  caps  in  the  plan  should  be  constructed  by  project- 
ing points  from  the  view  shown  in  the  side  elevation.  In 
actual  practice  in  the  drawing  room,  three  or  four  of  the 
principal  points  (those  that  mark  the  limits)  would  be  located 
and  the  curves  sketched  in  freehand,  they  being  inked  in  on 
the  tracing  by  aid  of  an  irregular  curve.  In  such  cases  the 
draftsman  has  a  good  idea  of  the  shape  of  the  curves,  owing 
to  previous  practice  in  drawing  them.  When  drawing  in 
the  curves  formed  by  the  opening  in  the  bedplate  shown  in 
this  view,  the  student  must  exercise  his  own  judgment 
regarding  their  shape,  taking  care  not  to  get  them  too 
straight.  The  general  drawing  gives  a  good  idea  of  their 
proper  curvature. 

Returning  to  the  elevation,  draw  the  crank  and  crank  end 
of  the  connecting-rod  in  the  position  shown  in  the  general 
drawing.  With  the  center  of  the  rrankpiii  as  a  center  and 
a  radius  equal  to  the  length  of  the  connecting-rod  between 
its  centers  (obtained  from  the  detail  sketch),  describe  an 
arc  cutting  the  center  line  pq  at  a  point  that  will  be  the 
center  of  the  crosshead  pin.  Draw  the  crosshead,  obtaining 
the  dimensions  from  the  detail  sketch.  Complete  the  con- 
necting-rod in  both  views  and  draw  the  piston  rcxl  1  inch  in 
diameter.  Draw  both  views  of  the  cylinder  with  the  nuts 
aiid  the  steam  pipe  in  their  proper  position,  getting  all 
dimensions  from  the  detail  sketches. 

Draw  the  center  line  of  the  valve  stem  in  the  plan  view, 
and  draw  the  stuffingbox.  valve  stem,  valve-stem  slide  and 
its  guide  in  both  views.  In  order  to  determine  the  position 
of  the  valve-stem  slide,  it  is  necessary  to  locate  the  center 
of  the  eccentric.  Referring  to  the  general  drawing,  it  is 
seen  that  the  eccentric  is  on  the  dead  center  farthest  from 
the  cylinder.  The  offset  of  the  eccentric  is  given  as  ^  inch 
in  the  detail  sketch ;  hence,  when  in  this  position,  the  center 
of  the  eccentric  strap  will  be  situated  |^  inch  to  the  right  of 
the  crank-shaft  center  on  the  line/f/.     With  this  point  as  a 


S6  MECHANICAL  DRATVIXG  ?  l-l 

center  and  a  radius  equal  to  the  distance  between  the  cen- 
ters of  the  eccentric  strap  and  the  hole  in  the  stub  end  of 
the  eccentric  rod  (see  detail  sketch),  in  this  case  2  feet 
i^  inches,  describe  an  arc  cutting  the  center  line  />q  in  O: 
O  will  be  the  center  of  the  pin  on  the  valve-stem  slide,  which 
may  be  completed  by  aid  of  the  detail  sketch.  Complete 
the  drawing  of  the  eccentric,  eccentric  strap,  and  eccentric 
rod  in  both  views. 

Finally,  draw  in  the  bandwheel  and  flywheel  (see  general 
drawing  for  position).  The  flywheel  will  be  of  the  same 
diameter  as  the  bandwheel,  but  only  3  inches  wide. 

The  pencil  drawing  is  now  completed.  Before  beginning 
to  trace,  erase  the  lines  that  are  not  to  be  inked  in.  This 
is  not  necessary,  but  it  is  better  to  do  so,  since  it  avoids 
confusion  and  lessens  the  liability  of  making  mistakes. 
Some  draftsmen  prefer  to  redraw  a  portion  of  those  parts 
that  are  to  be  inked  in  with  a  somewhat  softer  pencil  and 
leave  the  light  construction  lines  on  the  drawing  rather 
than  erase  them ;  in  some  cases,  this  saves  time. 

The  preliminary  directions  for  tracing  a  drawing  have 
been  given  previously.  First,  trace  the  side  elevation,  begin- 
ning with  the  flywheel,  and  then  as  much  of  the  connecting- 
rod,  eccentric,  and  eccentric  rod  as  can  be  seen.  Trace  all 
those  parts  of  the  bedplate,  cylinder,  valve  stem,  stuffing- 
box,  etc.  that  are  seen.  Then  trace  the  plan  view,  letter  the 
drawing,  and  draw  the  border  lines.  There  will  be  no  plate 
number  for  this  tracing,  but  the  student's  name,  class  num- 
ber, and  the  date  of  completion  will  be  put  on  as  before. 

The  student  should  exercise  particular  care  to  have  every 
iimension  scale  exactlv  the  size  given  in  the  detail  sketches. 


PRACTICAL   PROJECTION 


IXTROBUCTIOX 

1.  Ortliograpliie  Projection. — When  the  mechanic 
is  required  to  make  any  article  whose  form  or  dimensions 
are  not  previously  known,  it  is  evident  that  a  description 
of  the  work  in  question  should  be  furnished  him.  This 
description  may  be,  and  often  is,  given  by  verbal  instruc- 
tion; but  in  order  to  enable  the  worker  to  understand 
definitely  what  is  wanted,  the  form  of  the  object,  its  dimen- 
sions, and  the  quality  of  the  material  to  be  used  should  be 
stated.  Instruction  given  in  this  way  is,  however,  seldom 
satisfactory  either  to  the  workman  or  to  his  employer,  since 
it  is  difficult  in  such  cases  to  place  the  responsibility  for  any 
errors  that  may  occur. 

Written  instruction,  therefore,  would  seem  to  be  prefer- 
able; but,  since  most  objects  would  require  an  extended 
description,  a  shorter  and  more  convenient  method  of  con- 
veying the  desired  information  is  to  be  sought.  A  draAv- 
ing  of  the  object  is  therefore  made. 

These  drawings  are  generally  made  by  a  process  termed 
orthographic  projection,  or,  as  it  is  usually  called,  projec- 
tion druAving.  Every  detail  of  the  object  is  correctly 
represented  in  this  drawing,  so  that  the  workman  knowing 
how  to  "read  the  drawing"  may  obtain  his  measurements 
therefrom  for  the  construction  of  the  object  itself.  He  is 
also  enabled  by  an  examination  of  the  drawing  to  under- 
stand exactly  how  the  object  will  appear  when  completed. 
Hence,  we  have  the  following  definition: 

§15 

For  notice  of  copyright,  see  page  immediately  following  the  title  page. 


2  PRACTICAL    PROJECTION  §15 

Ortliogriiphic  pio.K  itioii  is  the  process  of  viaking  cor- 
rect representations  of  objects  by  iiteans  of  drawings.  ' 

3.  A  Working  Dra\rtiig  (Generally  Xecessary. — 
Before  a  pattern  for  any  article  can  be  made,  a  working 
drawing  is  needed.  Xo  pattern,  however  simple  or  plain,  can 
be  produced  until  we  have  something  definite  to  work  from. 

The  metal  worker  does  not  go  to  the  trouble  of  preparing 
a  drawing  on  paper  for  every  piece  of  work  he  is  called  on 
to  make,  since  many  objects  are  so  plain  that  a  brief  verbal 
or  written  description  of  their  dimensions  gives  the  mechanic 
all  the  information  he  needs  to  enable  him  to  lay  out  his 
work. 

If,  for  example,  a  tinsmith  is  called  on  to  make  a  box  out 
of  IX  tin,  -t  inches  long,  3  inches  wide,  and  1  inch  deep,  he 
immediately  proceeds  with  the  steel  square  to  lay  off  the 
given  sizes  directly  on  the  metal;  but  if  the  same  mechanic 
is  required  to  make  a  round  pan  having  flaring  sides  or  some 
article  of  a  form  not  readily  carried  in  the  mind,  there  is 
one  thing  he  must  do  before  he  can  proceed  with  the  work 
or  even  lay  out  the  pattern — ^he  must  make  a  working 
dra^vlng  of  the  object. 

3.  Wliat  Constitutes  a  Working  Drawing. — There 
are  several  ways  in  which  this  drawing  may  be  made, 
depending  altogether  on  how  complicated  the  object  is.  In 
the  case  of  the  pan  referred  to,  it  may  be  desirable,  by  an 
application  of  certain  principles,  to  omit  the  operation  of 
making  a  drawing  consisting  of  several  views  and  proceed 
as  Avith  the  box  to  '"lay  it  out"  directly  on  the  metal.  In 
this  case,  however,  it  will  be  found  that  the  operation 
differs  from  that  of  making  the  box  referred  to,  since  it  is 
first  necessary  to  mark  out  the  sizes  and  outlines  as  they 
will  appear  when  the  pan  is  completed.  These  sizes  may  or 
may  not  form  a  part  of  the  pattern,  but  thej'  are  required 
as  preliminary  lines  from  which  to  'May  off,"  or  ''strike 
out,"  the  pattern. 

Marking  out  these  sizes  or  dimensions  of  an  object  is 
really  making  a  working  drawing.     This  drawing  may  be 


§  15  PRACTICAL    PROJECTION  3 

full  size — in  which  case  it  is  referred  to  as  a  detail  dra^w- 

Ing — or  it  may  be  drawn  to  a  scale,  either  lart^er  or  smaller 
than  the  object  itself. 

4.  AVhere  the  l^ra^viiis'  Is^  Made, — In  the  case  of 
plain  articles,  the  necessary  drawings  may  be  made  directly 
on  the  metal;  in  the  majority  of  cases,  however,  the  work  is 
of  a  more  or  less  complex  nature,  makini;-  it  ]ii_L;hly  impor- 
tant that  a  full-sized  properly  made  detail  drawing-  be  used. 
This  is  not  always  provided,  and  the  mechanic  frequently 
has  to  make  his  own  detail  from  a  small  freehand  sketch  or 
possibly  from  a  drawing  made  to  a  small  scale.  In  the  latter 
case,  an  enlarged  drawing  must  generally  be  made  before 
the  work  of  laying  out  the  pattern  can  proceed.  This  neces- 
sitates operations  with  drafting  board  and  drawing  imple- 
ments— with  which  the  student  is  already  familiar.  The 
proficiency  that  has  been  acquired  may  now  be  put  into 
practical  use  in  the  operations  to  follow. 

It  is  the  purpose  of  this  section  to  present  methods  by 
which  working  drawings  may  be  made  and  read.  These 
methods  are  presented  in  a  practical  way,  that  the  principles 
laid  down  may  be  readily  understood  by  the  student. 


ge:n^eeal  principles 

5.  Various  Kinds  of  DraAvinjys. — The  most  common 
representations  of  objects  are  those  used  for  purposes  of 
illustration  merely  and  known  as  perspeetive  dra^vings. 
They  are  of  little  value  to  the  mechanic  to  serve  as  working 
drawings,  since  they  are  not  drawn  to  a  scale  in  the  same 
way  as  a  projection  drawing,  and  to  obtain  measurements 
therefrom  is  an  operation  both  complicated  and  indirect.  A 
photograph  is  an  ideal  perspective  picture,  but  no  one  would 
think  of  using  a  photograph  as  a  working  drawing.  The 
photograph  and  the  perspective  drawing  represent  the  object 
"  as  we  see  it,"  or  as  it  appears  to  the  eye  of  the  observer, 
while  the  working  drawing — the  projection  drawing — repre- 
sents the  object    as  it  actually  is,  or  will  be  when  made. 


4  PRACTICAL    PROJECTION  §  15 

A  photograph,  however,  shows  only  such  objects  as  really 
exist,  while  a  projection  drawing  often  shows  objects  that 
exist  only  in  the  imagination  of  the  draftsman  or  a  person 
capable  of  understanding,  or  "reading,"  the  drawing.  By 
means  of  such  drawings  the  imagination  is  aided  in  pictur- 
ing the  object  as  already  constructed — or,  as  we  would  say, 
it  enables  the  mechanic  to  see  the  object  "in  his  mind's  eye." 

6.  "WTiat  Is  SlioAvn  In  a  Drawing. — The  perspec- 
tive drawing  always  shows  more  than  one  side  of  an  object — 
generally  three  sides — while  the  working  drawing  seldom 
shows  more  than  one  side,  that  being  the  side  towards  the 
observer.  The  position  of  such  other  portions  of  the  object 
as  are  not  located  on  the  side  shown  in  the  drawing  may, 
however,  be  indicated  in  a  projection  drawing  by  dotted 
lines.  Xo  lines  should  be  used  in  a  working  drawing  that 
do  not  represent  actual  edges,  or  outlines,  in  the  object 
itself.  We  sometimes  find  certain  edges  or  outlines  of  an 
object  represented  in  a  drawing  by  heavy  lines.  These 
heavy  lines  are  called  shade  lines ;  but  since  they  are  not 
essential  features  of  the  working  drawing,  no  description  of 
them  is  necessary  in  this  section.  There  is  also  an  elaborate 
system  of  representing  the  effect  of  light  and  shade  on 
curved  and  receding  surfaces  by  means  of  lines  properly 
disposed  over  the  surfaces  shown  in  the  drawing.  Since 
these  lines  are  for  effect  only,  and  their  meaning  is  apparent 
to  the  observer,  the  principles  governing  their  use  are  not 
made  a  part  of  the  subject  matter  of  this  section. 

7.  Position  of  Observer.  —  Another  point  of  differ- 
ence between  the  perspective  and  the  projection  drawing  is 
that,  in  taking  the  viev.^  of  which  the  perspective  drawing 
is  a  representation,  the  eye  of  the  observer  remains  in  a 
fixed  position,  and  in  the  same  relation  to  the  drawing  as  the 
camera  is  to  the  photograph ;  while  in  that  view  of  the 
object  of  which  the  projection  draAving  is  a  representation, 
the  eye  of  the  observer  is  ahvays  supposed  to  be  directly 
over  or  opposite  that  point  in  the  drawing  which  is  being 
noted. 


§15 


PRACTICAL    PROJECTION 


This  may  be  illustrated  by  the  student  for  himself  in  a 
very  simple  way.  Place  a  sheet  of  paper  on  the  drawing- 
board  and  draw  a  horizontal  line,  say  (J  inches  long;  now 
lay  an  ordinary  2-foot  pocket  rule  against  this  line,  in  the 
manner  shown  in  Fig.  1,  and   proceed  to  mark  off  the  line 


Fig.  1 

to  correspond  with  the  divisions  on  the  rule.  He  will  find 
that  he  is  obliged  to  get  his  eye  directly  over  each  mark  on 
the  rule,  and  to  "sight"  carefully  down  on  to  the  rule 
before  making  each  mark,  very  much  as  he  would  "sight" 
or  look  along  a  piece  of  work  to  see  whether  it  is  straight  or 
not.  It  will  be  noticed  also  that  he  is  making  otic  eye  do 
all  the  work  in  this  "sighting,"  and,  further,  it  will  be 
observed  that  in  making  the  markings  he  is  moving  his  head 
as  he  progresses  towards  the  end  of  the  line.  He  is  obliged 
to  do  this  to  keep  his  eye  exactly  over  each  point  on  the  line 
as  it  is  marked. 

8,  liine  of  8iglit. — The  line  first  drawn  on  the  paper 
is  not  the  only  one  made  use  of  in  reproducing  the  markings 
on  the  rule.  The  student  has  unconsciously  made  use  of 
another  line,  or,  more  properly,  a  set  of  lines,  that  arc  an 


6  PRACTICAL    PROJECTION  §  15 

important  feature  of  projection  drawing.  These  lines  are 
those  made  use  of  in  doing  the  "sighting"  necessary  for 
the  marks ;  they  are  purely  imaginary  lines  and  are  not  rep- 
resented in  the  illustration.  They  are  very  properly  called 
lines  of  sight.  The  lines  of  sight  in  a  projection  drawing 
are  ahi^'ays  pcrpendiciilar  to  the  draiving.  They  extend  from 
a  point  in  the  eye  of  the  observer  to  a  point  on  the  drawing 
that  is  directly  opposite,  as  indicated  by  the  point  of  the 
draftsman's  pencil  in  Fig.  1. 

These  lines  of  sight — which,  as  stated,  are  only  imaginary 
lines  and  are  not  represented  in  a  working  drawing — con- 
stitute one  of  the  most  important  features  of  the  projection 
drawing;  for  on  these  lines  we  are  enabled  to  obtain  the 
views  from  the  object,  and,  by  means  of  other  lines,  called 
projectors,  bearing  a  certain  relation  to  the  lines  of  sight 
(as  will  be  explained  later),  Ave  can  reproduce  the  views  thus 
obtained  on  the  drawing. 

9.  Xines  of  8iglit  Always  Parallel.  —  When  it  is 
desired  to  make  a  drawing  of  any  object,  the  lines  of  sight 
must  be  used  in  the  same  manner  as  in  marking  the  divisions 
on  the  line  in  Fig.  1 ;  that  is,  care  must  be  taken  to  keep 
the  lines  of  sight  in  any  one  view  parallel  to  one  another. 
We  may  take  different  views  of  the  same  object,  or,  to 
express  it  otherwise,  we  may  take  positions  on  different 
sides  of  the  object,  in  order  to  obtain  views  therefrom;  but 
in  any  view  thus  taken  the  above  statement  must  be  care- 
fully observed  and  the  lines  of  sight  kept  exactly  parallel  to 
one  another. 

10.  Several  A'iews  Xecessarj'.  —  AVe  have  already 
noted  that  a  projection  drawing  seldom  shows  but  one  side 
of  an  object.  Since  there  are  no  objects  that  present  all 
their  dimensions  on  any  one  side,  it  necessarily  follows  that, 
in  order  to  convey  a  correct  idea  of  the  form  of  an  object, 
it  is  necessary  to  make  a  drawing — or  a  projection — of  as 
many  sides  as  will  enable  the  correct  shape  and  dimensions 
to  be  shown.  We  may  make  these  drawings  from  as  many 
points  of  view  as  may  be  desired ;  but  for  certain  reasons, 


§  15  PRACTICAL    PROJECTION  7 

to  whirh  attention  will  l)e  called  later,  it  is  t^cnerally  prefer- 
able to  view  all  objects  from  six  sides,  which  correspond  to 
the  six  sides  of  a  cube. 

11.  Before  we  proceed  with  the  explanation  of  ortho- 
graphic projection,  it  is  important  that  the  student  should 
be  informed  about  what  are  known  as  the  aiigifs  of  pro- 
jection. That  is  to  say,  he  must  know  something  about  the 
positions  supposed  to  be  assumed  by  the  object  when  the 
different  views  are  taken.  Draftsmen  generally  recognize 
for  projection  drawing  two  principal  angles  in  which  the 
objects  are  placed,  and  the  drawings  made  in  these  two 
angles  are  called,  respectively,  Jirst-aitgic  projcctioiis  and 
tJiird-aiigIc  projections. 

These  different  positions  of  the  object  will  be  understood 
by  the  student  from  an  examination  of  Fig.  2,  in  which  four 
angles  are  formed  by  the  intersection  of  the  horizontal 
plane  y^  B  C D\n\\\\  the  vertical  plane  E FG  H.  Then,  of  the 
four  right  angles  included  between  the  planes,  the  angle  B PG 
is  called  the  first  angle,  the  angle  GPC  \'~>  known  as  the 
second  angle,  the  angle  C  PF  is  the  third  angle,  and  the 
angle  F P B  is  the  fourth  angle.  In  the  view  shown  at  {a) 
these  two  planes  are  assumed  to  be  opaque  and  the  object  to 
be  transparent.  The  lines  of  sight  are  projected  from  the  eye 
of  the  observer  to  the  plane  of  projection,  tJtrough  the  object. 
In  the  view  at  (b)  the  planes  are  assumed  to  be  transparent 
and  the  lines  of  sight  pass  from  the  object  to  the  eye  of  the 
observer,  through  the  plane.  The  effect  of  these  different 
positions  of  the  object  is  merely  to  change  the  relative  posi- 
tions of  the  different  views  when  made  on  the  drawing  board. 

The  drawings  made  in  this  section  are  first-angle  pro- 
jections. Third-angle  projections  are  made  by  processes 
similar  to  those  here  explained  and  the  different  positions  of 
the  views  will  be  pointed  out  to  the  student  at  the  proper 
time. 

13,  Plans  and  Elevations. — It  being  assumed  tliat 
the  object  is  in  some  fixed  position,  the  various  views  take 
their  names  from   the  different  positions  of  the  observer  in 


PRACTICAL    PROJECTION 


15 


his  vie^y  of  the  object.      Thus,  a  view  taken  from  above,  or 
looking  down  on  the  object,  is  called  a  plan ;    so  also  is  a 

H  I 


view  from  beneath,   or  looking  up  at   the   object;  thus  we 
have  the  terms  top  plan  and  bottom  plan.      The  two  views 


§  15  PRACTICAL    PROJECTION  9 

thus  obtained  are  frequently  desio-nated  by  terms  that  vary 
with  the  chiss  of  objects  represented  and  not  infrequently 
derive  their  names  from  some  portion  of  the  object  itself. 
Thus,  a  top  i)lan  of  a  house  is  a  view  of  the  roof  taken  from 
above  and  is  called  a  roof  plan  ;  while  a  ceiling  plan  is,  as 
its  name  indicates,  a  view  of  another  part  of  the  house  taken 
from  the  opposite  direction.  In  the  case  of  small  objects 
generally,  such  views  are  termed  top  plan  or  bottom  plan,  as 
the  case  may  be.  These  views,  in  certain  cases,  should  be 
marked  on  the  drawing,  in  order  to  guard  against  error. 
Here  it  should  be  noted  that  while  the  position  of  the  object 
is  not  changed  in  making  either  the  top  or  the  bottom  plan, 
yet  the  position  of  the  observer  is.  When  the  two  i)lans 
thus  made  are  compared,  it  is  found  that  the  corresponding 
points  of  the  draAvings  are  changed  in  their  relation  to  each 
other  in  the  same  manner  as  the  hands  of  two  persons  that 
are  standing  exactly  in  front  of  and  facing  each  other— the 
right  hand  of  the  one  being  opposite  to  the  left  hand  of  the 
other. 

A  vievv'  taken  from  the  side  of  an  object  is  called  an 
elevation.  That  side  of  an  object  shown  in  any  elevation 
gives  its  name  to  that  drawing;  thus,  a  view  of  the  front  of 
an  object  is  called  a  front  elevation.  So,  also,  we  have  the 
terms  rear  elevation  and  side  elevation.  In  some  cases  it  may 
be  more  convenient  to  designate  the  elevations  by  the  points 
of  the  compass;  for  example,  the  north  elevation  of  a  build- 
ing is  a  projection  of  that  part  of  the  building  which  faces 
north,  or,  to  state  it  as  we  have  done  before,  that  part  of 
the  building  seen  when  looked  at  from  the  north. 

13.  Section  Drawings. — Cases  frequently  occur  in 
which  the  views  or  dimensions  desired  to  be  given  on  a 
working  drawing  cannot  be  shown  in  either  plans  or  eleva- 
tions. Under  such  circumstances,  recourse  is  often  had  to 
a  class  of  drawings  termed  sections.  A  section  drawing  is 
a  projection  of  an  object  assumed  to  have  been  cut  in  two 
in  a  certain  direction,  usually  at  right  angles  to  the  lines  of 
sight.      Those  parts  of  the  object  between  the  observer  and 


10  PRACTICAL    PROJECTION  §  15 

the  place  where  the  cut  is  made  are  assumed  to  have  been 
removed,  so  as  to  present  an  entirely  new  surface.  This 
surface  is  not  seen  in  the  object  itself,  since  the  cutting  is 
entirely  imaginary — done  simply  for  the  purpose  of  showing 
some  interior  construction. 

The  cut  just  referred  to  may  be  made  in  a  horizontal,  a 
vertical,  or  an  oblique  direction,  according  to  the  way  in  which 
it  is  desired  to  show  the  section.  Portions  of  surfaces  that 
have  been  cut  in  this  way  are  usually  represented  by  certain 
conventional  methods,  indicating  the  character  or  composition 
of  the  material.  A  custom  frequently  adopted,  and  which 
will  be  followed  in  the  drawings  for  this  section,  consists  in 
designating  surfaces  exposed  by  the  cut  by  a  series  of  closely 
drawn  parallel  lines.  Such  lines  are  usually  drawn  at  an 
oblique  angle,  as  compared  with  the  other  portions  of  the 
drawing,  and  are  called  cross-section  lines  or  cross-hatching. 

14.  A  Set  of  Plans. — It  is  a  common  practice,  when 
speaking  of  a  set  of  drawings  consisting  of  various  plans, 
elevations,  sections,  etc. — as  for  a  house  or  for  some  other 
object — to  refer  to  them  as  a  "set  of  plans."  This  is  a 
collective  phrase  for  the  drawings  and  its  use  in  this  way  is 
perfectly  proper ;  when  used  in  this  sense,  however,  it  is  not 
understood  as  applying  simply  to  a  plan  view  as  explained 
in  Art.  12.  Drawings  for  large  objects  are  frequently  of  a 
size  such  that  the  different  views  are  more  conveniently 
made  on  separate  sheets.  Architectural  drawings  are  usually 
separated  in  this  way,  and  it  is  often  necessary  for  the  one 
that  is  to  read  such  drawings  to  arrange  the  sheets  in  a 
particular  manner,  in  order  that  the  relation  between  the 
views  may  be  understood.  This  arrangement  of  the  views 
will  be  considered  later. 

15.  Foresliortene<l  View*;. — The  lines  in  a  projection 

drawing — or.  as  we  shall  term  it  hereafter,  a  projection — are 
either  of  the  same  length  as  the  corresponding  edges  or  out- 
lines of  the  object  itself,  or  they  are  foreshortened.  No 
lines  are  represented  longer  than  they  actually  exist  on  the 


§  15  PRACTICAL    PROJECTION  11 

object,  except  in  cases  where  the  drawing-  is  made  to  an 
enlarged  scale.  All  lines  that  are  used  to  represent  the 
edges  or  outlines  of  an  object,  and  that  are  at  right  angles  to 
the  lines  of  sight  in  any  view,  are  represented  in  that  view 
by  their  true  length.  Lines  that  represent  other  edges  or 
outlines,  and  are  not  at  right  angles  to  the  lines  of  sight,  are 
consequently  represented  shorter  than  they  exist  on  the 
object,  and  are  then  said  to  be  foreshortened. 

16.  Geometrical  Forms. — The  simplest  geometrical 
form  that  we  can  imagine  is  ^  point;  next  we  have  a. /iui\ 
defined  as  the  shortest  distance  between  two  points;  then  a 
surface,  which  is  a  flat,  or  plane,  figure  bounded  by  lines ;  and, 
finally,  a  so/i(/,  which  in  turn  is  bounded  by  different  surfaces. 

17.  Tlie  Combination  of  Geometrical  Forms. — Since 
the  mechanic  deals  with  objects  of  various  forms  that 
may  be  said  to  represent  geometrical  solids,  we  shall 
endeavor  to  convey  an  idea  of  the  way  in  which  the  repre- 
sentation of  these  objects  may  be  simplified  by  resolving 
them  into  their  elements,  or  the  parts  that  combine  to  pro- 
duce these  forms.  We  thus  have  to  deal  with  points,  lines, 
surfaces,  and  solids;  of  these  four  things,  one  only — the 
solid — can  be  represented  by  any  actual  thing  or  be  such 
as  to  enable  us  to  handle  it,  for  there  is  no  object  that  does 
not  possess  IcngtJi,  brcadt/i,  and  thickness  to  a  greater  or  less 
extent.  It  consequently  folUnvs  that  the  other  three  forms 
are  entirely  imaginary.  The  student  that  can  most  readily 
conceive  or  imagine  their  existence  in  this  way  will  most 
readily  comprehend  the  principles  involved  in  projection 
drawing  and  pattern  drafting. 

18.  A    Test    of    the    Student's    Imajjrination.  —  The 

following  illustration  Avill  show  how  a  solid  may  be  resolved 
into  the  simplest  of  its  elements  and  still  retain  its  definite 
and  characteristic  form.  We  will  suppose  a  sheet-metal  box 
to  be  made  in  the  form  of  a  cube,  each  edge  being  1  inch 
long.  The  six  square  pieces  of  sheet  metal  that  make  the 
sides  of  this  box  arc  t(^  be  lightly  soldered  together,   "edge 


12  PRACTICAL    PROJECTION  §  15 

and  edge. "  This  box  represents  a  geometrical  solid,  although 
it  is  by  no  means  a  solid  considered  in  a  physical,  or  prac- 
tical, sense.  It  is,  in  fact,  popularly  spoken  of  as  being 
"hollow  ";  but  we  could  very  readily  convert  it  into  a  solid 
by  filling  it  with  molten  solder.  It  represents,  however,  as 
it  is,  a  geometrical  solid,  and  as  such  we  Avill  consider  the 
parts  of  which  it  is  made  up,  without  paying  any  attention 
in  this  elementary  part  of  the  subject  to  the  material  of 
which  it  is  composed.  "We  are  now  dealing  with  forms  only, 
and  until  these  principles  are  fixed  in  the  student's  mind  no 
attention  will  be  paid  to  other  details. 

19.  We  first  look  for  the  surfaces  of  this  solid,  of  which 
we  find  six.  The  student  may  have  some  difficulty  in  under- 
standing that  the  surfaces  of  this  cube,  which  he  can  appar- 
ently distinguish  by  the  sense  of  touch,  exist  only  in  an 
imaginary  way.  We  refer  him,  therefore,  to  our  definitions 
for  the  explanation  of  this  seeming  paradox.  A  surface,  we 
have  been  told,  has  length  and  breadth,  while  a  solid  has 
these  and  also  a  third  property — thickness.  Now,  as  we 
have  said  before,  we  cannot  consider  the  material  of  which 
the  cube  is  composed,  but  if  we  handle  it  we  will  be  obliged 
to  refer  to  the  metal  of  which  the  sides  of  the  cube  are  made. 
These  metal  sides  have  thickness;  it  may  be  only  a  few 
thousandths  of  an  inch,  but  still  it  is  thickness,  and  conse- 
quently does  not  come  within  our  definition  of  a  surface.  A 
surface  may  be  compared  to  a  shadow,  which  can  be  dis- 
tinguished by  its  outlines  or  shape,  but,  as  every  one  knows, 
is  absolutely  without  thickness.  It  is  in  this  sense  that  we 
refer  to  the  sides  of  the  cube  as  its  surfaces.  We  find  also 
that  each  of  these  six  surfaces  is  bounded,  or  defined,  by  four 
edges  or  outlines,  the  lines  in  turn  being  terminated  at  the 
corners  of  the  cube  by  points,  of  which  there  are  eight.  It 
will  now  be  seen  that  these  eight  points,  situated  at  the 
extremities  of  the  lines,  or  edges,  of  the  cube,  perfectly 
define  the  shape  and  size  of  the  geometrical  solid. 

If  we  could  imagine  the  metal  sides  of  the  cube  to  dis- 
appear entirely,  leaving  only  the   points  at  the  corners,  we 


§  15  PRACTICAL    PROJECTION  13 

would  still  have  as  perfect  a  representation  of  the  size  and 
shape  of  the  cube  in  our  minds  as  if  it  had  an  actual  exist- 
ence and  could  be  seen  by  the  eye.  For  these  imaginary 
points  could  then  very  easily  be  imagined  as  being  connected 
by  lines,  and  we  could  then  "  see  "  the  surfaces,  and  finally 
the  solid,  existing  in  the  same  imaginary  way.  It  will  thus 
be  comparatively  easy  for  us  to  transfer  a  representation  of 
this  solid  to  our  drawing,  since  we  can  use  the  points  as  the 
markings  on  the  rule  were  used  in  connection  with  the  lines 
of  sight  in  a  previous  illustration. 

20,     The    liuaj>'inati()ii   ii   ^■^llllJl^)le    Assistant   to   tlie 

Draftsman.  —  x\n  object,  or  solid,  of  any  conceivable 
shape  may  thus  be  resolved  into  its  elementary  parts  or 
points.  The  drawing  of  the  object,  then,  will  consist  simply 
of  locating  the  positions  of  these  points  on  the  drawing. 
We  may  have  drawings  to  make  that  will  require  the  location 
of  a  hundred  or  more  of  these  points,  depending  entirely  on 
the  form  or  shape  of  the  object  we  are  dealing  with,  but  the 
principles  are  in  all  cases  the  same. 

If  the  student,  after  resolving  an  object  in  this  imaginary 
way,  will  carefully  study  or  imagine  the  proper  location  of 
these  points  in  their  relation  to  the  object  itself,  defining 
their  positions  on  the  drawing  one  at  a  time,  much  that  may 
appear  complicated  at  first  sight  will  resolve  itself  into  very 
simple  and  comparatively  elementary  work.  Complicated 
work  is  usually  nothing  more  or  less  than  the  aggregation  of 
a  number  of  simjjle  operations  that  appear  complicated  only 
because  they  are  combined.  There  is  no  field  of  work  to 
which  the  latter  statement  is  more  applieable  than  to  that  of 
the  draftsman. 

It  is  in  the  "imaginary"  way  thus  described  that  the 
student  is  directed  to  picture  to  himself  each  figure  as  pre- 
sented tt)  him  for  the  making  of  the  drawings  on  the  plates. 
This  part  of  the  study  is,  as  will  be  noticed,  almost  entirely 
the  work  of  the  imagination ;  but  it  should  be  practiced  by 
the  student  for  the  sake  of  the  assistance  it  will  be  to  him 
later  on. 

M.  E.    v.— IS 


14  PRACTICAL    PROJECTION  §  15 

The  operations  of  projection  drawing  follow  one  another 
in  a  natural  sequence,  which  we  will  proceed  to  trace  out  in 
a  series  of  drawing  plates.  As  the  student  follows  these 
operations,  keeping  in  mind  the  foregoing  principles,  he  will 
have  no  difficulty  in  making  or  reading  any  drawing. 


PLATES 

21,  Seven  plates  are  to  be  drawn  by  the  student  in 
accordance  with  the  directions  given  in  this  section.  They 
are  to  be  of  the  same  size  as  those  drawn  for  Geometrical 
Draining,  and  the  same  general  instructions  regarding  the 
preparation  of  the  plates  are  to  be  observed;  they  must  be 
drawn  and  sent  to  us  for  correction  in  the  same  manner. 

The  letter  heading  for  each  problem,  which  has  heretofore 
been  placed  on  the  drawing,  will  be  omitted,  and  the  stu- 
dent is  required  only  to  designate  each  plate  with  the  letter 
heading,  or  title,  that  is  printed  in  heavy-faced  type,  both 
in  this  section  and  on  the  reduced  copies  of  the  plate.  For 
this  purpose  the  block-letter  alphabet  is  used. 

22.  The  dimension  lines  and  figures  shown  in  the  first 
three  problems  of  the  drawing  plate,  title:  Projections  I.  are 

to  be  especially  noticed  by  the 


Right 


Wrong— - 


student.  They  are  ordinarily 
used  in  all  working  draw- 
ings, and  preference  is  invari- 
ably given  to  a  dimension 
^"^  ^  figure,  rather  than  to  the  scale 

to  which  a  drawing  is  made.  Dimension  figures  are  not  to 
be  placed  on  the  plates,  since  the  object  in  requiring  the 
student  to  draw  these  projections  is  rather  to  enable  him  to 
gain  an  idea  of  their  principles  than  to  be  able  to  make  a 
finished  working  drawing. 

Dimension  and  extension  lines  when  used  should  be  light 
broken  lines.  Care  should  be  exercised  to  make  the  arrow- 
heads as  neatly  as  possible  and  of  a  uniform  size — not  too  flar- 
ing. They  are  made  with  a  steel  writing  pen  and  their  points 
should  touch  the  extension  lines,  as  illustrated  in  Fig.  3. 


VP 


1 


E^- 


^F         t£ 


.J_X. 


PRDJEfl 


.1 L. 


PROBLEMS. 


"T"i — r  \—^ 

X 


■-f- 


'     I. 


t- 


PROBLEM  S.  I 


+ 


PROBLEM  6. 


JUNE  25, 1893. 


Copyright,  1899,  by  The  Cc 
All  righi 


riDNS-I. 


WLEMS. 


>-  PROBLEMS, 


,-i 


_      _      '.  x\  i 
K    AH'    \ 


/      57 

PROBLEM  7.    \° 


PROBLEMS. 


ERV  Engineer  Company. 
iserved. 


JO/iA/  ^Af/T/y,  CLA22  A/?  45S9. 


§  15  PRACTICAL    PROJECTION  15 

DRAAVING  PLATE,   IIILE:    PROJECTIONS  I 

33.  General  Instructions. — This  plate  is  divided  into 
four  equal  spaces,  and  each  of  these  divisions,  with  the 
exception  of  the  upper  left-hand  space,  is  again  divided,  by- 
means  of  a  central  vertical  liiie,  into  two  equal  parts.  Use 
light  pencil  lines,  as  they  are  not  to  be  inked  in  and  are 
intended  only  to  facilitate  the  location  of  the  problems. 
These  lines  are  not  shown  on  the  printed  copies  of  the 
plates.  Before  attempting  to  draw  any  of  the  problems 
on  the  plate,  the  explanations  accompanying  each  problem 
should  first  be  carefully  read  and  compared  with  the;  reduced 
copy  of  the  plate  and  also  Avith  the  illustrations  in  this  sec- 
tion. The  principles  of  projection  drawing  will  thus  be 
better  understood  by  the  student  and  their  application 
readily  made  when  more  difficult  drawings  are  undertaken. 

TIic  fiDidamcntal  lazvs  of  projection  are  contained  in  the 
first  four  probleihs,  and  if  these  are  thoroughly  mastered  by 
the  student,  the  application  of  the  lazes  to  the  reuiaifiing prob- 
lems zvill  be  comparatively  easy. 

34.  Why  Different  VieAvs  Are  Drawn. — The  names 
of  the  different  views  have  already  been  noted;  in  this 
plate  it  is  shown  how  they  are  distinguished  from  one 
another  in  a  drawing.  The  relation  of  the  different  views 
to  one  another  will  also  be  exjjlained. 

Some  objects  in  certain  positions  may  have  all  their 
dimensions  represented  in  two  views — a  i)lan  and  an  eleva- 
tion— but  generally  three  views  should  be  drawn.  There 
are,  indeed,  many  cases  where  views  from  each  of  the 
six  sides,  as  well  as  sections  and  views  taken  from  oblique 
positions,  are  necessary.  It  has  already  been  observed  that 
lines  are  represented  in  their  true  length  only  when  at  right 
angles  to  the  lines  of  sight;  consequently,  since  the  posi- 
tion of  the  object  in  any  set  of  views  is  not  changed,  it  is 
necessary  to  change  the  i)osition  of  the  observer  in  such  a 
manner  as  to  bring  the  lines  of  sight  where  they  will  be  at 
right  angles  to  the  lines  in  the  object,  thus  enabling  the 
latter  to  be  shown  in  their  true. length  on  the  drawing. 


16  PRACTICAL    PROJECTION  §  15 

The  student  will  readily  perceive  that  in  drafting-room 
work  it  is  of  the  highest  importance  that  the  lines  which 
combine  to  make  up  the  surfaces  of  any  object  should  be 
shown  in  their  true  length,  or  at  least  be  presented  in  such 
positions  that  their  true  lengths  may  be  easily  found.  With- 
out these  true  lengths,  no  measurements  can  be  obtained 
from  which  to  lay  out  patterns,  a  pattern  being  merely  a 
representation  of  the  surfaces  of  some  solid.  It  is  neces- 
sary, therefore,  to  be  prepared  to  take  views  of  any  object 
from  any  position;  for  there  are  many  different  forms,  or 
shapes,  of  solids,  and  it  is  necessary  to  be  able  to  show  in  its 
true  length  any  line  in  the  object  that  may  be  needed  for  a 
pattern. 

25.  The  Base  Line. — We  shall  first  consider  objects 
in  positions  that  may  be  shown  in  two  views.  Draw  a 
horizontal  line  through  the  central  portion  of  the  upper  left- 
hand  space  on  the  drawing,  as  at  tn-n  on  the  plate.  In  the 
portion  of  the  space  below  this  line  are  to  be  drawn  the  top 
plans  of  each  of  the  two  simplest  forms,  viz. ,  the  point  and 
the  line.  The  space  above  this  line  is  to  contain  the  eleva- 
tions of  the  same  forms.  The  line  thus  drawn  is  called  a 
Itase  line  and  defines  the  boundary  of  the  surfaces  on 
which  we  are  to  "sight,"  or,  as  we  shall  say  hereafter,  on 
which  we  are  to  project  the  lines  of  sight.  It  is  necessary 
to  call  the  imagination  into  use  again  and  imagine  the 
paper  to  be  bent  up  at  a  right  angle  on  this  line. 

26.  Plane<  of  Projection.  —  The  drawing  paper  is 
imagined  to  be  the  surface  that  intercepts  the  lines  of  sight; 
and  in  the  case  of  a  plan,  as  seen  from  instruction  already 
given,  must  be  a  horizontal  surface,  while  in  the  case  of  an 
elevation,  it  is  imagined  to  be  a  vertical  surface.  The  dif- 
ferent portions  of  the  drawing  on  which  the  projections  are 
made  are  called  planes  of  projection,  and  are  also  distin- 
guished by  other  names  that  designate  the  position  they  are 
supposed  to  occupy  in  intercepting  the  lines  of  sight.  That 
portion  of  the  drawing  on  which  the  elevation  is  drawn  is 
called  the  vertical  plane  of  projection;    it  is  represented 


§  15  PRACTICAL    PROJECTION  17 

on  this  plate  by  the  space  above  the  base  line;  the  portion 
below  the  base  line  is  devoted  to  the  plan  and  is  called  the 
horizontal  plane  of  projection.  We  shall,  for  the  sake  of 
brevity,  refer  to  these  surfaces  by  the  use  of  the  letters 
V  P  and  H  P,  respectively.  Copy  these  letters  into  the  upper 
and  the  lower  left-hand  portion  of  their  respective  spaces 
on  the  drawing,  using  for  that  purpose  a  block  letter  one- 
half  the  size  of  the  title  letter  and  leaving  a  distance  of 
^  inch  from  the  border  lines  of  the  spaces. 

27.  Foot  of  tlie  Tjine  of  Siglit. — Before  proceeding 
with  the  drawing  of  this  plate,  it  is  desired  to  call  the  atten- 
tion of  the  student  to  the  distinction  to  be  observed  between 
the  ii)iaginative  and  the  practical  features  of  this  subject. 
The  imaginative  feature  is  employed  when  a  conception  of 
an  object  is  formed  by  the  student  in  accordance  with  the 
instruction  in  previous  articles,  and  also  when  the  lines  of 
sight  are  applied  in  the  imaginary  way,  as  in  the  "  sighting  " 
illustrated  in  Fig.  1.  The  application  of  the  practical 
feature  in  this  instance  is  made  when  the  position  of  each 
division  of  the  rule  is  indicated  on  the  drawing  by  a  pencil 
mark  or  dot.  The  practical  part  of  the  work  is  always 
accomplished  by  the  aid  of  pencil  and  drawing  instruments. 

The  two  features  are  closely  associated,  since  we  cannot 
have  a  practical  representation  of  any  object  without  first 
having  an  idea,  or  an  imaginative  conception,  either  of  the 
object  itself  or  of  the  means  of  projection.  The  practical 
feature  of  the  work  was  introduced  in  the  illustration  (Fig.  1) 
when  a  mark  or  dot  was  made  on  the  paper,  thereby  indica- 
ting the  position  of  the  point  at  which  the  line  of  sight  was 
intercepted  by  the  plane  of  projection. 

That  point  on  any  plane  or  drawing  where  a  line  of  sight 
is  intercepted  is  called  the  foot  of  the  line  of  sight.  When 
the  foot  of  every  line  of  siglit  that  can  be  used  on  the  ele- 
mentary points  of  any  object  is  thus  represented  on  the 
drawing  by  dots,  and  connecting  lines  are  drawn  between 
such  dots,  the  drawing  is  completed,  and  the  object  is  said 
to  be  "■  projected." 


18  PRACTICAL    PROJECTION  §  15 

28.  Projectors. — If  but  one  view  of  an  object  were 
required,  the  use  of  the  lines  of  sight  as  previously  explained 
(representing  the  imaginative  feature)  and  the  drawing  of 
the  dots  and  lines  referred  to  in  the  previous  article  (repre- 
senting the  practical  feature)  would  be  all  that  is  necessary 
for  the  student  to  understand  before  proceeding  with  the 
work  on  the  drawing  board. 

Since  it  has  been  shown  that  several  views  are  required, 
another  important  practical  feature  must  necessarily  be 
explained.  This  relates  to  the  connection  usually  established 
between  the  different  views  of  a  drawing  and  the  lines  that 
are  drawn  in  a  certain  manner  between  corresponding  points 
in  each  view.  These  lines  are  usually  not  represented  in  a 
finished  drawing,  since  they  are  in  the  nature  of  construction 
lines.  They  are  essential,  however,  to  the  work  of  making 
the  drawing,  and  it  is  very  important  that  the  student  should 
thoroughly  understand  the  principles  by  which  they  are 
employed.  These  lines  are  called  projectoi-s  and  may  be 
defined  as  the  trace  of  a  line  of  sight,  or  the  representation 
of  the  foot  of  a  line  of  sight  moving  in  a  certain  direction. 
Projectors  are  used  in  two  ways,  which  are  distinguished 
from  each  other  for  the  present  by  the  terms  primary  and 
secondary. 

29.  Primary  Projectors. — This  use  of  projectors  is 
illustrated  in  Fig.  1,  which  shows  the  drawing  bent  up  at  a 
right  angle  along  the  base  line  vi-n.  The  point  A  is  pro- 
jected to  HP  by  the  vertical  line  of  sight  C  B\  it  is  also 
projected  to  V  P  by  the  horizontal  line  of  sight  D  E\  B  and  E 
are  dots  at  the  foot  of  each  line  of  sight. 

It  is  assumed  that  the  first  position  of  the  observer  is  at  C\ 
he  then  moves,  in  the  direction  of  the  arrow,  along  the 
dotted  line  to  D.  If.  in  so  doing,  he  continues  to  sight 
through  the  point  J.  it  is  apparent  that  a  line  will  be  traced 
from  j5  to  /'on  H  P  and  from  F  Xo  E  oxi\  P.  The  upright 
portion  of  the  drawing  V  P  is  now  imagined  to  be  bent  back- 
wards until  laid  flat  on  the  drawing  board,  and  it  is  evident 
that  E  F  B   is    represented  on  the  flat  surface  of  the  two 


15 


PRACTICAL    PROJECTION 


19 


planes  of  projection  by  the  strai^lit  line  Jl'  F  P>.  It  may 
therefore  be  drawn  as  a  straight  line  by  the  aid  of  the  tri- 
angle and  T  square,  the  position  of  the  point  .  /  in  each  view 
being  determined  by  the  points  B  and  E'  at  the  extremities 


mtST.  POSITION 


Fig.  4 


of  the  line.  These  points  (or  dots,  for  points,  being  entirely 
imaginary,  could  not,  of  course,  be  actually  represented)  * 
B  and  £'  are  the  projections  of  the  point  A — the  line 
drawn  between  them(/>7^^')  is  called  2,  projector.     When 


*  Attention  has  been  called  to  the  fact  that  jioints,  lines,  and  surfaces 
are  entirely  imaginary  geometrical  forms.  This  is  true  in  the  sense 
that  the  student  must  consider  such  forms  in  the  imaginative  study  of 
this  subject.  When  their  representation  on  the  drawing  paper  is  con- 
sidered, however,  something  that  can  actually  be  seen  by  the  eye  is 
required.  Therefore,  when  a  point  is  referred  to  in  this  section  as  per- 
taining to  the  drawing,  it  is  to  be  represented  by  a  neat  dot  in  the 
proper  place  on  the  paper.  In  like  manner,  a  line  should  be  represented 
by  a  fine  pencil  mark  drawn  between  two  points  marking  its  extrem- 
ities. When  the  line  is  to  serve  a  special  purjxjse,  as  e.xplained  in  this 
section,  it  is  inked  in  in  a  particular  manner  characteristic  of  its  use,  in 
order  that  the  drawing  may  be  more  easily  read.  A  surface,  therefore, 
would  be  represented  by  a  portion  of  the  drawing  bounded  by  the 
proper  lines  and  descrijitive  of  the  form  of  surface  rei)resentcd.  Re- 
member that  accurate  work  cannot  be  done  unless  the  pencil  points  are 
in  good  condition.  The  student  should  provide  himself  with  a  smooth 
file  or  piece  of  fine  sandpaper  and  frequently  sharpen  the  chisel  point 
of  the  leads  in  both  pencil  and  compasses,  in  order  that  fine  sharp  lines 
may  be  readily  drawn. 


20  PRACTICAL    PROJECTION  §  15 

projectors  are  used  as  in  this  illustration — that  is,  between 
two  planes  that  may  actually  be  bent  up  as  shown  in  Fig.  4 — 
they  are  said  to  be  used  in  a  primary  manner. 

The  secondary  use  of  the  projector  will  be  shown  in  con- 
nection with  Problem  3,  Case  III. 

The  practical  use  of  the  projector  is  clearly  shown  in  the 
following  problems.  It  is  a  most  important  factor  in  the 
projection,  and,  as  will  be  seen  from  instruction  soon  to 
follow,  is  often  the  first  line  to  be  used  in  a  drawing. 


PROBLEM   1 

30.  To  project  tlie  plan  and  elevation  of  an  iniag- 
inai^  point. 

There  are  two  cases  of  this  problem,  representing  differ- 
ent positions  of  the  point.  Definite  instructions  are  given 
for  drawing  the  first  projection  and  the  student  is  expected 
to  draw  the  second  projection  without  further  directions. 

Case  I. —  WJien  the  point  is  located  1  inch  from  each  of  the 
two  surfaces  V  V  and  W  P. 

This  position  is  illustrated  in  Fig.  4,  referred  to  in 
Art.  29. 

CoxsTRUCTiox. — Fix  a  point  B  (see  plate)  1  inch  below 
the  base  line  on  the  drawing.  This  point  should  be  f  inch 
from  the  left-hand  side  of  the  drawing  and  is  the  plan  view 
of  the  point  given  in  the  problem.  Bring  the  T  square  and 
triangle  into  position  and  draw  the  projector  vertically 
upwards  and  across  the  base  line.  Fix  a  point  2i\..E'  on  the 
projector  1  inch  above  the  base  line.  A  projection  drawing 
is  thus  made,  showing  two  views — a  plan  and  an  elevation 
— of  the  required  point,  the  position  of  which  is  thus  defi- 
nitely established. 

Fig.  4  is  an  illustration  of  the  imaginative  feature  and 
the  projection  drawing  just  made  is  a  representation  of  the 
practical  feature  of  the  work — the  part  actually  made  by 
the  draftsman.  The  intimate  connection  between  the  two 
features  may  be  seen  if  the  drawing  just  made  is  compared 


§  15  PRACTICAL    PROJECTION  31 

with  the  illustration  in  Fig.  4.  Similar  results  are  found 
to  have  been  accomplished  in  both  cases,  the  method  last 
employed  being  the  only  one  practicable  for  actual  use. 
When  inking-  in  this  drawing,  make  small  round  dots  to 
represent  the  positions  of  the  points,  and  always  ink  in 
projectors  as  hght  dot-and-dash  lines,  as  shown  on  the 
plate.  These  dot-and-dash  lines  should  be  inked  in  in  a 
uniform  manner,  as  on  the  plate,  the  dashes  being  about 
i  inch  in  length  and  spaced  about  ^V  inch  apart,  with  a 
light  dot  between  each  dash.  Measure  the  distances  by  the 
eye  and  preserve  a  uniform  shade  for  all  projectors,  thus 
giving  the  drawing  a  neat  appearance.  The  base  line  should 
be  represented  by  a  heavy  dotted  line,  as  shown  at  in-n  in 
the  perspective  illustration  of  Fig.  4. 

When  making  the  preliminary  drazvings,  do  not  attempt 
to  drazu  dotted  lines  wit/i  the  peneil,  sinee  this  is  liable  to 
affect  the  aeeuracy  of  the  zvork.  Keep  the  chisel  point  of 
the  pencil  sharp  and  draw  as  fine  a  line  as  can  be  distinctly 
seen.  The  contrast  between  the  different  lines  on  the  draw- 
ing may  then  be  clearly  indicated  when  the  work  is  inked  in. 

Case  H. —  ]]'hen  the  point  is  located  Ih  inches  from  V  P 
and  ^  inch  from  H  P. 

The  student  will  fix  the  location  of  the  point  in  the  plan 
and  elevation  on  the  drawing  without  further  instructions, 
bearing  in  mind  the  fact  that  distances  from  V  P  are  meas- 
ured on  the  plan  and  distances  from  H  P  are  shown  in  the 
elevation.  Reference  to  Fig.  4  explains  this  statement. 
Case  II  should  be  placed  on  the  drawing  about  I-  inch  to  the 
right  of  the  preceding  figure. 


PROBLEM  3 

31.  To  project  the  plan  and  elevation  of  an  imagr- 
iuary  line,  the  line  being  in  a  rifjfht  position. 

The  term  right  position  is  used  in  connection  with  pro- 
jection drawing  as  distinguished  from  the  terms  inclined, 
or  oblique,  position.      The  line,  therefore,  can   be  either  in  a 


22 


PRACTICAL    PROJECTIOX 


§15 


horizontal  position  or  in  a  vertical  position  and  still  be 
designated  as  in  a  rigJit  position.  There  are  three  figures 
for  this  problem,  representing  three  cases  Avhere  the  line  is 
in  a  right  position  and  yet  represented  differently  on  the 
drawing.  The  different  positions,  the  various  distances, 
and  the  length  of  the  lines  for  the  three  cases  of  this  prob- 
lem are  clearly  illustrated  in  the  perspective  drawings 
shown  in  Figs.  5,  G,  and  7.  Instructions  are  given  for  the 
drawing  of  Case  I  on  the  plate,  but  the  student  is  expected 
to  be  able  to  make  the  drawings  for  Cases  II  and  III  with- 
out further  directions  than  those  contained  in  the  illustra- 
tions. Be  careful  to  preserve  a  distance  of  \  inch  between 
the  drawings,  so  that  the  plate  may  present  a  neat  appear- 
ance when  completed. 

Case  I. —  WJicn  the  line  is  parallel  to  both  H  P  and\  P. 

ExPLAXATioN. — This  position  of  the  line  is  illustrated  in 
Fig.   5,  and,  as   apparent  from  that  figure,  the  drawing  is 


Fig.  5 
merely  an  extension  of  Problem  1.  Each  end  of  the  line 
is  treated  as  a  point,  projected  first  to  the  plan  and  after- 
wards to  the  elevation  in  precisely  the  same  manner  as  was 
the  point  in  Problem  1,  the  only  difference  being  that  there 
are  two  points  instead  of  one,  for  we  cannot  have  a  line 
without  establishing  at  least  two  points. 


§  15  PRACTICAL    PROJECTION  23 

This  problem  also  illustniLes  another  principle  of  projec- 
tion already  referred  to,  viz.,  all  lines  at  ri;4ht  angles  to  the 
lines  of  sight  in  any  view  are  shown  in  that  view  in  their 
true  length;  or,  in  other  words,  the  lines  that  are  to  be 
made  on  the  drawing  to  represent  the  plan  and  elevation 
of  A  B,  Fig.  5,  will  be  of  the  same  length  as  A  B  is  indi- 
cated in  the  figure,  viz.,  2  inches.  The  angles  HA  B  and 
L  A  B  are  right  angles,  although  shown  in  perspective  in 
the  figure;  and,  since  the  lines  of  sight  in  any  view  are 
always  parallel  to  each  other,  the  angles  GBA  and  KB  A 
must  also  be  right  angles;  consequently,  as  the  line  A  B  is 
at  right  angles  to  the  lines  of  sight  in  both  views,  it  must 
be  shown  in  its  true  length  in  both  the  plan  and  the  eleva- 
tion on  the  drawing. 

Construction. — To  make  this  drawing  on  the  plate,  draw 
a  horizontal  line  of  the  given  length  and  the  proper  dis- 
tance (i.  e. ,  1  inch)  below  the  base  line.  This  will  be  the 
plan  of  the  line  ./  />'.  From  each  end  of  this  line  {CD  in 
Fig.  5  and  on  the  plate)  draw  projectors  to  the  elevation; 
or,  to  use  the  term  by  which  such  operations  are  desig- 
nated, project  the  ends  of  the  line  CD  to  the  elevation. 
After  measuring  off  the  proper  height  above  the  base  line, 
draw  the  horizontal  line  E  F,  which  is  the  elevation  of  the 
line  A  B. 

Note. — When  a  point  is  projected  from  one  view  to  another,  its 
projector  (a  straight  line)  is  drawn  from  the  first  view  to  the  view  pro- 
jected, and  always  at  right  angles  to  the  base  line. 

Case  II. —  Where  t lie  line  is  in  a  Jiorizontal  position  ODui 
at  ri^a^Jit  angles  to  ^  P. 

ExPL.\N.\Tiox. — Fig.  0  illustrates  this  case.  It  will  be 
noticed  that  the  plan  view  of  the  line  does  not  differ  very 
much  from  the  plan  of  the  line  given  in  Case  I,  merely  that 
it  is  represented  by  a  vertical  line  on  the  drawing  in  the 
plan  in  place  of  a  horizontal  line,  as  in  Case  I.  The 
line  ^  .5  is  at  right  angles  to  the  vertical  lines  of  sight  G  D 
and  H C  in  both  cases.  The  line  .'/  B  in  this  figure  is  in 
such  a  position  that  the  horizontal  line  of  sight  K£  passes 


24 


PRACTICAL    PROJECTION 


§15 


through  both  points  B  and  A.  The  projection  of  these 
points,  therefore,  on  the  elevation  is  the  single  point  E  at 
the  foot  of  the  line  of  sight  K E.  A  single  line  of  sight  may 
pass  through  an  unlimited  number  of  points  in  any  view, 
but  the  foot  of  such  line  of  sight  is  always  represented  in 
that  view  by  one  point  on  the  drawing.  The  student  who 
is  to  read  that  drawing  must  picture  to  himself,  or  imagine, 
the  position  of  these  points  as  they  are  supposed  to  exist  in 
the  object  of  which  any  drawing  is  a  representation.  It  is 
further  to  assist  his  imagination  that  other  views  are  drawn, 
and  by  means  of  which  the  position  of  the  different  points 
may  be  definitely  located.      Thus,  in  this  case,  if  we  were 


Fig.  6 
to  consider  the  elevation  alone,  without  paying  any  atten- 
tion to  the  plan,  we  Avould  say  that  the  point  E  represented 
merely  some  other  point  (imaginary,  of  course)  that  could 
be  situated  anywhere  on  the  line  of  sight  K E.  A  glance  at 
the  plan,  however,  shows  not  only  where  the  location  is, 
but  how  many  points  are  represented.  In  this  case  there 
are  two  points  represented  by  E,  one  directly  in  line  with 
the  other.  The  plan  also  shows  how  far  apart  these  points 
are,  and  from  the  two  views  it  can  be  further  seen  that  the 
line  is  in  a  right  position  perpendicular  to  V  P.  If  it  were 
not,  the  two  points  would  not  be  in  the  same  line  of  sight, 
and  would,  consequently,  require  two  positions  on  the  ele- 
vation, whereas  they  are  designated  by  the  one  position  E, 
Fig.  6. 


15 


PRACTICAL    PROJECTION 


25 


Construction.— Since  the  drawings  for  this  case  have 
been  shown  in  the  preceding-  exphination  to  be  similar  to 
those  of  Case  I,  no  definite  instructions  need  be  given.  The 
two  drawings  differ  in  position  only  and  are  drawn  as 
shown  on  the  plate,  tiie  plan  being  first  constructed. 

Case  III.  —  JJ7/cn'  the  line  is  in  a  vertieal position. 
This  is  shown  in  Fig.  7,  each  feature  of  which  has  already 
been   explained    in    connection    with    Figs.   5    and    G.     The 


Fig.  7 

drawing  may,  therefore,  be  made  by  the  student  in  accord- 
ance with  tlie  dimensions  given  in  the  illustration. 

32.  Proof  of  a  Projection  I)rawinjt»-.— The  various 
cases  of  the  foregoing  problem  represent  lines  indifferent 
positions,  and,  as  in  the  case  of  the  point  in  Problem  1,  the 
student  will  see  that  these  projections  definitely  repre'sent 
the  position  of  each  line;  further,  that  for  each  position 
indicated,  but  one  line  can  be  placed,  or  can  occupy  that 
position.  It  is  recommended  that  the  student  prove  this 
assertion  as  follows:  Copy  the  projections  of  this  problem 
on  another  piece  of  paper  and  bend  the  paper  at  right 
angles  along  the  base  line,  as  shown  in  tlu-  illustrations; 
take  a  piece  of  small  wire  of  the  given  length,  to  represent 
the  lines,  and  proceed  to  hold  it,  in  turn,  over  the  drawing 
for  each   case,  at    the    same  time  "  sighting,"  or  using  the 


26 


PRACTICAL    PROJECTION 


§15 


lines  of  sight,  as  illustrated.  It  will  be  seen  that,  in  order 
to  make  the  foot  of  each  separate  line  of  sight  come  to  the 
proper  place  on  the  drawing,  the  wire  must  be  held  in  the 
position  Indicated  in  the  statement  of  the  case. 


PKOBIJEM  3 

33.  To  draw  the  projections  of  an  iniasinary  line 
in  A  riglitlj-  melinecl  position- 
There  are  three  cases  of  this  problem,  in  all  of  which  the 
given  line  is  rightly  inclined;  that  is,  the  angle  of  inclina- 
tion is  such  that  the  true  length  of  the  line  may  be  shown 
in  either  a  plan,  a  front  elevation,  or  in  some  elevation  that 
shall  be  at  right  angles  to  the  front  elevation.  The  differ- 
ent cases  of  this  problem  are  presented  in  perspective  views, 
from  which  the  projection  drawings  are  to  be  made  by  the  stu- 
dent.    They  illustrate  the  principles  of  foreshortened  views. 

Case  I. — WJiere  tlu  line  is  horizontal,  but  inclined  t:  V  P 
at  an  angle  of  ^. 

ExPLAKATiojf. — This  is  shown  in  Fig.  8,  which  gives  all  the 
dimensions  and  distances  necessary  to  enable  the  student  to 


Fig.  8 


draw  the  projections  on  the  plate.     The  student  will  note  the 
position  of  each  point  and  carefully  observe  the  instructions 


§  15  PRACTICAL    PROJECTION  27. 

for  making  the  drawings.  Bear  in  mind  that,  although 
the  drawing  is  only  that  of  a  single  line,  careful  study  nurst 
be  given  to  it,  for  the  principles  on  which  these  simple  pro- 
jections are  made  are  the  same  as  for  any  other  projection 
drawing.  These  principles  are  shown  in  a  more  compre- 
hensive way  in  simple  problems  than  if  an  object  of  com- 
plex form  were  presented,  requiring  a  confusing  number  of 
points  to  define  its  outline.  In  Cases  I  and  II  of  this  prob- 
lem, the  plan  is  first  to  be  drawn  and  the  elevation  projected 
'therefrom. 

Construction.— Since  the  line  in  Case  I  is  in  a  horizontal 
position  and  therefore  at  right  angles  to  the  vertical  lines 
of  sight,  it  will  be  shown  in  its  full  length  on  the  plan.  The 
line  is  stated  to  be  inclined  to  V  P  at  an  angle  of  45°;  draw 
the  plan,  therefore,  at  that  angle  to  the  base  line  and  at 
such  distance  below  the  base  line  as  indicated  in  Fig.  8  and 
shown  at  A  B  on  the  plate.  Project  the  points  A  and  B  to 
the  elevation,  and  at  the  given  height  above  the  base  line 
draw  a  horizontal  line  between  the  projectors.  This  is  the 
elevation  of  the  line  shown  in  the  plan,  and  since  its  entire 
length  is  contained  between  the  horizontal  lines  of  sight  F E 
and  H  G,  Fig.  8,  the  line  cannot  be  shown  on  the  elevation 
as  being  any  longer  than  the  perpendicular  distance  between 
the  projectors.  This  distance  being  less  than  the  actual 
length  of  the  line,  the  elevation  is,  in  this  case,  called  a 
foreshortened  view  of  the  line.  It  represents,  however,  the 
entire  line,  and  reference  to  the  plan  is  necessary  in  order  to 
find  its  true  length. 

C&,se.  11.— Where  the  line  is  parallel  to  V  P  but   inelined 
to  HP  at  an  angle  of  60°. 

Construction.— Fig.  9  shows  that  the  plan  is  to  be  rep- 
resented by  the  horizontal  line  F If.  It  also  gives  the  length 
of  the  line  in  the  i)lan,  which  is  a  foreshortened  view. 
Therefore,  draw  F H  1  inch  long  and  1  inch  below  the  base 
line.  Draw  the  projectors  and  fix  a  point  at  /  on  the  pro- 
jector drawn  from  /%  at  the  proper  distance  above  the  base 
line.     This  will   be  one   end  of    the   line   in   the   elevation 


28 


PRACTICAL    PROJECTION 


§15 


With  the  compasses  set  to  2  inches  (the  length  of  A  B)  and 
using  the  point  y,  fixed  on  the  projector  drawn  from  i%  as  a 
center,  describe  an  arc  intersecting  the  other  projector. 
The  point  that  represents  the  other  end  of  the  Hne  is 
located  at  this  intersection,  and  the  line  may  then  be  drawn. 
Now  bring  the   T  square   into  position  and  prove  by  the 


Fig.  9 

triangle  that  the  line  y  A' is  at  an  angle  of  G0°  with  the  base 
line.  It  will  be  seen  that  it  would  have  been  possible,  after 
fixing  the  position  of  a  point  at  either  end.  to  have  drawn 
the  line  at  once  with  the  60°  triangle.  Attention  is  called 
to  both  methods  in  order  to  show  the  student  the  connection 
between  them. 

Case  III. —  ]Vhc}'c  the  line  is  rigJitly  inclined  to  V  P  at  an 
angle  of  6'6'°  and  is  also  inclined  to  H  P. 

ExPLAXATiox. — In  Cases  I  and  II,  the  line  has  been  in 
such  positions  as  to  enable  its  full  length  to  have  been 
shown  in  one  of  the  views  drawn  on  the  plate.  In  this  case, 
a  position  is  illustrated  in  which  the  line  is  shown  foreshort- 
ened in  both  of  these  views.  It  will  therefore  require 
another  view  to  be  projected  in  order  that  the  line  may'be 
shown  in  its  true  length.  Since  it  is  known  that  the  line  is 
rightly  inclined,  the  additional  view  required  will  be  at  right 


15 


PRACTICAL    PROJECTION 


29 


angles  to  the  base  line.  As  another  view  is  to  be  drawn,  so 
another  base  line  will  be  required.  This  base  line,  being 
merely  the  lower  boundary  of  a  surface  supposed  to  be  in  an 
upright  position  to  receive  the  lines  of  sight  and  at  right 
angles  to  the  surface  of  the  elevation  previously  drawn,  will, 
consequently,  be  at  right  angles  with  the  base  line  in  a 
drawing  that  shows  only  two  views — such  drawings  as  have 
thus  far  been  made. 

Fig.  10  contains  the  given  dimensions,  etc.  for  this  position 
of  the  line.      This  perspective  figure  shows  the  interception 


Fig.  10 

of  the  lines  of  sight  from  still  another  direction  than  has 
been  shown  in  the  preceding  illustrations.  In  the  same 
manner  a  view  may  be  obtained  from  any  side  of  an  object 
or  at  any  angle  other  than  a  right  angle.  The  method  of 
accomplishing  these  results  by  the  use  of  the  T  square  and 
triangle  on  the  flat  surface  of  the  drawing  board  will  now 
be  shown,  and  the  illustration  of  the  bent-up  surfaces  will 
not  be  continued  beyond  this  problem.  Such  illustrations 
are,  however,  always  implied  in  a  projection  drawing,  for 
that  part  of  the  work  is  the  imaginative  feature  previously 
mentioned,  to  which  the  attention  of  the  student  will  be 
directed  throughout  this  instruction. 

Since  the  angle  of  inclination  to  the  elevation  is  the  same 

in  this  case  as  in  the  plan  of  Case   II,  the  length   shown  on 

the  elevation  of   this   projection  will  be  the  same  as  in  the 

plan  of  that  case.      In  this  drawing,  the  line  is,  however,  in  a 

M.  E.     v.— 14 


30  PRACTICAL    PROJECTION  §  15 

different  position  as  related  to  the  plan,  and  from  Fig.  10  it 
will  be  seen  that  it  must  be  represented  by  a  vertical  line  in 
that  view  on  the  drawing. 

CoxsTRUCTiON. — Extend  the  oase  line  on  the  plate  to  the 
center  of  the  next  space,  from  which  point  draw  a  vertical 
line  downwards  to  the  division  line;  these  lines  are  to  be 
inked  in  the  same  as  the  base  line  in  the  first  space.  In  the 
smaller  space  thus  enclosed  on  the  drawing  is  to  be  drawn 
the  plan  for  this  problem,  the  front  elevation  occupying  the 
same  relative  position  as  before,  directly  above  the  plan,' 
while  in  the  space  at  the  right  the  side  elevation  will  be 
projected.  First,  draw  the  elevation  as  at  iT '  F'  on  the  plate, 
fixing  the  point  E '  at  the  specified  distance  above  the  base 
line.  As  the  foreshortened  length  in  the  plan  is  given  in 
Fig.  10  as  If  inches,  draw  the  line  C  D  oi  that  length,  as 
shown  on  the  plate,  keeping  the  point  C  at  its  proper  dis- 
"tance  below  the  base  line,  as  indicated  in  Fig.  10.  The  view 
to  represent  this  line  in  its  true  length  may  next  be  drawn. 
It  is  known  that  this  view  must  be  one  in  which  the  line 
itself  is  represented  at  right  angles  to  the  lines  of  sight. 
There  is  a  choice  of  two  views  for  this  projection — either  to 
the  right  or  to  the  left  side.  Having  already  utilized  the 
space  to  the  left  on  the  plate,  the  side  elevation  is,  in  this 
case,  projected  to  the  right.  The  method  employed  in 
Case  II  might  here  be  used  to  project  the  side  elevation,  but 
since  it  is  customary,  when  a  number  of  elevations  are  pro- 
jected from  the  same  plan,  to  facilitate  the  operation  by 
drawing  between  such  views  lines  that  are  termed  second- 
ary projectors,  an  explanation  of  their  use  is  here  presented. 

34.  Secondary  Use  of  the  Projector.  —  The  term 
secondary  is  not  applied  in  the  case  of  projectors  as  indica- 
ting an  unimportant  or  infrequent  application  of  these  lines. 
The  name  is  used  rather  to  distinguish  operations  in  which 
similar  principles  as  applied  to  the  imaginative  features  are 
differently  represented  on  the  drawing  in  the  application  of 
the  practical  features  of  the  work.  In  fact,  both  uses  of 
these  lines  are  required  in  most  drawings.      It  is  therefore 


§15 


PRACTICAL    PROJECTION 


31 


essential  that  the  student  should  become  familiar  with  the 
various  means  employed  in  producing  them  on  the  drawing. 

35.  It  is  evident,  from  an  inspection  of  Fig.  10,  that  the 
eye  of  the  observer  at  E  in  moving  around  along  the  broken 
line  in  the  direction  of  the  arrows  to  take  position  at  K 
would  trace  a  line  from  F,  through  O,  to  /..  The  part  of 
this  line  {F O)  that  shows  on  the  front  elevation  is  parallel 
to  the  base  line  of  that  surface;  and,  also,  that  part  of  the 
line  {O  L)  shown  on  the  side  elevation  is  parallel  to  the  base 
line  of  the  side  -elevation.  It  is  seen  that  the  definition  of 
the  projector,  as  previously  given,  applies  equally  to  the 
lines  F  O  and  O  L.  If  the  two  planes  of  projection  repre- 
sented by  the  two  upright  surfaces  could  be  bent  in  the 
same  relation  to  each  other  as  were  the  plan  and  the  elevation 
in  Fig.  4 — i.  e.,  on  the  line  P  Q,  Fig.  10 — the  use  of  the  pro- 
jectors in  these  views  would  be  no  different  from  that  already 
described.  It  is  customary,  however,  in  first-angle  pro- 
jection, to  assume  that  such'upright  surfaces  are  always  bent 
downwards  and  away  from  the  plan ;  to  accomplish  this 
result,  the  secondary  use  of  the  projector  is  employed.  Sup- 
pose, now,  that  the  upright  surfaces,  represented  in  Fig.  10, 
were  bent  backwards  until  laid  flat  on  the  drawing  board. 
Evidently,  there  would  be  an  appearance  presented  similar 
to  that  shown  in  Fig.  11,  and  an  open  space  would  be  shown 


Fig.  11 


on  the  drawing  board  included  between  the  angle /* (7/*'. 
The  paper  on  which  the  drawing  is  made  is  not  of  an  irregular 
shape,  thus  to  be  bent  up  at  will;  further,    the  operations 


3-2  PRACTICAL    PROJECTIOX  §  15 

performed  in  projection  drawing  are  such  that  they  can  be 
accomplished  only  on  the  flat  surface  of  the  drawing  board. 

36.  It  is  found  that  similar  results  may  be  obtained  in 
two  wavs,  both  being  easily  affected  by  the  aid  of  the  draw- 
ing instruments.  The  first  is  known  as  the  angular  method, 
and  is  thus  accomplished:  If  the  projectors  F  O  and  O'  L, 
Fig.  11.  or  any  other  corresponding  set  of  projectors,  parallel 
to  their  respective  base  lines,  are  extended  until  they  inter- 
sect each  other,  it  is  found  that  all  the  intersections  arc  on  a 
diagonal  line  terminating  exactly  at  the  intersection  of  the 
base  lines.  It  is  also  found  that  this  diagonal  line  exactly 
bisects  the  outer  angle  formed  by  the  base  lines.  Applying 
these  principles,  therefore,  to  the  drawing,  bisect  the  outer 
angle  formed  by  the  base  lines  on  the  plate  and  produce  the 
bisector  indefinitely  towards  the  right-hand  side  of  the  space. 
The  outer  angle  formed  by  the  base  lines  in  this  case  being 
an  angle  of  270',  the  bisector  ma/  be  drawn  with  the  45" 
triangle,  since  a  line  thus  drawn  will  be  at  an  angle  of  135° 
with  both  base  lines. 

3T.     Fig-.   1"2  is  a  reproduction  of  the  projection  drawing 
from   the  plate,  showing  the  bisector  drawn  as   previously 
sEcosx,AMT  ivojxcrom     /directed.     Draw    F' x   and   F' y 
^/  i     parallel    to  the  base  line    in  the 
front  elevation;  from  their  inter- 
sections  with   the   bisector   at    x 
and  r,  draw  x  L  and  y  R  parallel 
to  the  base  line  in  the  side  eleva- 
tion.    Project  the  points  C  and  D 
from  the  plan  to  the  side  elevation 
by  the  use  of  primary  projectors, 
as     previousl)'    described.       The 
side  elevation  of  the  line  A  B  of 
Fig.   10,  then,  is  a  line  drawn  be- 
tween   points   of   intersection    of 
^^-  ^~  the    primary   with   the   secondary 

projectors,  as  shown  by  R  L,  Fig.  1-2.     The  lines  F'  x.  x  L. 
and  E '  y,  y  R  are  called  secondary  projectors,  and  are  used, 


§  15  TRACTICAL    PROJECTION  33 

as  in  this  case,  Avhen  projectinij  points  between  views  that 
are  related  to  one  another  in  the  manner  shown.  Project- 
ors are  used  in  a  similar  way  when,  for  reasons  that  will  be 
shown  later,  the  base  lines  are  at  an  angle  other  than  a 
right  angle.  In  all  cases,  the  outer  angle  is  bisected  as 
shown  in  Fig.  12,  and  the  secondary  projectors  are  drawn 
parallel  to  their  respective  base  lines. 

Note  that  tJic  front  elevation,  s//07^'u  in  Fig.  12,  is  a  fore- 
shortened fieza,  and  corresponds  in  length  with  t  lie  perpen- 
dicular distance  between  the  secondary  projectors  in  the 
side  elevation. 

38.  Readinjjf  a  Drawing. — The  ability  to  read  a  draw- 
ing consists  of  the  intelligent  comparison  of  the  different 
views  and  is  well  illustrated  in  the  projections  just  drawn. 
The  different  views — or  the  different  projections,  as  they 
are  called — must  never  be  considered  as  drawings  apart 
from  one  another.  Each  projection  is  shown  to  be  neces- 
sary in  order  to  enable  the  position  of  some  point  or  element 
of  the  object  to  be  established  in  the  reader's  imagination. 


PROBLEM  4 

39.  To  draAv  the  projections  of  an  imaginary  line 
in  an  obliquely  inclined  position. 

The  projections  of  this  problem  are  to  be  drawn  by  the 
student  in  the  next  space  on  the  plate,  following  the  instruc- 
tions here  given. 

Construction-. — Draw  the  base  lines  as  in  the  last  space 
used  for  Problem  3,  but  place  the  lines  \  inch  higher  on  the 
plate  and  extend  them  h  inch  farther  to  the  right  in  the 
space.  These  base  lines  will  be  used  in  the  construction  of 
the  projections  as  before,  but  will  not  be  inked  in  on  this 
drawing ;  they  are  construction  lines  onl)^ — to  be  erased  from 
the  plate  after  the  drawing  is  completed.  It  has  been 
shown  that  base  lines  are  necessary  for  determining  the 
position    of    the    different    points    on    a  drawing    and    are 


34 


PRACTICAL    PROTECTION 


S  15 


essential  in  establishing  the  first  few  points  in  any  projec- 
tion; but  as  the  drawing  progresses  and  other  lines  are 
produced,  any  right  line  in  a  view — i.  e.,  a  line  at  right 
angles  to  the  lines  of  sight — may  be  used  as  a  base  from 
which  to  establish  the  position  of  points  in  a  drawing. 


FIG.  13 

Represent  a  foreshortened  view  of  this  line  in  the  plan  by 
a  line  li^  inches  long,  drawn  at  an  angle  of  45°  with  the  base 
line  of  the  front  elevation,  as  shown  at  A  B.  Fig.  13;  draw 
the  front  elevation  (also  a  foreshortened  view)  at  an  angle 
of  30°  with  the  base  line.      Draw  the  line  A    B  in   the   plan 


§  15  PRACTICAL    PROJECTION  35 

and  G  E  in  the  front  elevation  in  such  positions  that  the 
end  of  either  Hne  nearest  to  the  base  Hue  shall  be  k  inch 
from  that  line.  Next  draw  the  side  elevation  as  explained 
in  Problem  3,  Case  III,  and  it  will  be  seen,  when  the  side 
elevation  is  completed,  that  H  J  \s  also  a  foreshortened 
view,  not  representing  the  true  length  of  the  line. 

An  elevation  will  now  be  projected  in  which  the  line  may  be 
shown  in  its  true  length.  This  will  be  an  elevation  whose 
surface  is  parallel  to  the  line.  Draw  the  base  line  of  this 
surface  ^  inch  from  the  line  A  B  on  the  plan  and  parallel  to 
that  line,  as  at  C  D,  Fig.  13.  This  figure  is  an  illustration 
of  the  projection  drawing,  showing  all  the  lines  used  in 
its  construction,  certain  of  which,  as  already  explained, 
are  not  to  appear  in  the  completed  drawing  on  the  plate. 
Note  that  the  oblique  elevation  K  L  is  projected  in  the 
same  manner  as  the  side  elevation  was  drawn,  the  only 
difference  being  that  the  outer  angle  O  D  C,  formed  by  the 
base  lines,  is  greater  than  a  right  angle,  but  is  treated  in 
the  same  way.  This  completes  the  problem,  and  in  finish- 
ing the  figure  on  the  plate,  the  student  will  ink  in  only  the 
different  views  and  the  primary  projectors,  erasing  all  other 
construction  lines. 

4:0.     Finding    True    Ijengfths    by   Triangles.  —  It    is 

possible  to  find  the  true  lengths  of  lines  from  a  plan  and  any 
elevation    showing    such    lines   obliquely  ^ 

inclined   by  a    shorter  method  than  that 
given  in  Problem  4.     This  is  accomplished 
by  the  use  of  the  right-angled  triangle.  ^' 
If  such  a  triangle  is  constructed,  with  its  ^'*"  '^ 

base  equal  to  the  length  of  the  line  shown  in  the  plan  and 
its  altitude  equal  to  the  vertical  height  shown  in  the  eleva- 
tion, t/ie  Jiypotciiusc  li'iH  be  equal  to  tJie  ty-ue  length  of  the 
line.  This  is  shown  in  Fig.  14,  in  which  A  B  is  made  the 
same  length  as  A  B,  Fig.  13,  and  B  C\  Fig.  14,  is  equal  to 
the  vertical  height  shown  in  the  elevation,  i.  e.,  E  F,  Fig.  13. 
Since  A  B  T  is  a  right  angle,  the  hypotenuse  A  C,  Fig.  14, 
is  equal  to  the  true  length  of  the  line.     This  statement  is  of 


36  PRACTICAL    PROJECTION  §  15 

the  greatest  importance  to  the  draftsman  and  should  be 
proved  by  the  student.  Construct  a  triangle  on  a  separate 
piece  of  paper  and  set  off  the  lengths  from  the  drawing  with 
the  dividers ;  afterwards  compare  the  length  of  the  hypot- 
enuse with  the  length  of  the  line  shown  in  the  oblique  eleva- 
tion, or  full  view.  This  is  an  illustration  of  a  principle  of- 
much  use  in  later  problems,  and  one  on  which  certain  impor- 
tant principles  of  patterncutting  depend. 

41.  All  Projections  Depend  on  Similar  Prin- 
ciples.— There  is  no  conceivable  position  of  a  line  that  may 
not  be  shown  or  its  true  length  not  be  ascertained  by  the 
application  of  the  principles  contained  in  the  foregoing 
simple  problems.  Lines  have  been  used  to  illustrate  these 
problems  drawn  at  such  angles  as  were  conveniently  made 
with  the  T  square  and  the  45°  or  60°  triangles,  but  any  angle 
or  any  position  could  as  well  have  been  represented,  since  the 
principles  are  in  any  and  all  cases  the  same.  "We  will  now 
proceed  with  the  representation  of  flat,  or  plane,  surfaces. 

42.  Planes,  or  Plane  Surfaces. — All  drawings  made 
to  represent  surfaces  are  composed  of  lines  that  bound,  or 
limit,  their  borders,  or  sides.  These  drawings,  therefore, 
will  differ  from  those  of  the  foregoing  problems  only  in  the 
fact  that  they  are  the  representation  of  lines  shown  in  their 
relation  to  one  another.  There  are,  however,  certain  prin- 
ciples relating  to  flat,  or  plane,  surfaces  that  must  be  borne 
in  mind,  since  they  influence  this  relation  of  the  different 
lines  in  a  drawing. 

43.  That  the  student  may  have  a  thorough  knowledge 
of  the  principles  employed  in  the  representation  of  surfaces, 
it  is  essential  that  he  first  have  a  clear  conception  of  what  a 
plane  is.  A  plane  surface,  as  has  been  stated,  has  only  an 
imaginary  existence,  being  bounded,  or  enclosed,  by  imagi- 
nary lines ;  this  surface  may  be  in  any  conceivable  position,' 
but  is  always  a  flat  surface.  If  viewed  from  a  certain  direc- 
tion— viz.,  as  if  "on  edge" — it  would  be  represented  by  a 
single  straight  line.  If  the  student  can  imagine  a  plane 
surface  indefinitelv  extended  in  everv  direction  bevond  the 


§  15  PRACTICAL    PROJECTION  37 

boundary  lines  of  the  figure,  he  will  have  a  very  good  con- 
ception of  a  plane;  any  number  of  points  or  lines,  the  posi- 
tions of  which  are  anywhere  on  this  surface  thus  extended, 
are  said  to  be  "  in  the  same  plane  "  in  relation  to  one  another. 

44.  To  illustrate:  Suppose  two  flat-top  tables  of  the 
same  height  are  on  the  floor  of  a  room  perfectly  level  and  of 
indefinite  extent.  Here  is  a  practical  representation  of  two 
planes,  both  of  them  in  a  horizontal  position;  one  plane  is 
represented  by  the  floor,  while  the  other  plane  is  parallel  to 
the  first  and  "passes  through  "  the  tops  of  the  tables.  The 
surfaces  represented  by  the  tops  of  the  tables  are  said  to  be 
"in  the  same  plane."  The  tables  may  be  placed  some 
distance  apart,  yet  the  straight  edge  of  a  ruler  laid  across 
their  tops  would  exactly  coincide  Avith  the  upper  surfaces  of 
both  tables  and  would  remain  in  contact  at  all  points  for 
every  position  of  the  ruler. 

The  plane  surface  represented  by  the  top  of  one  table  is 
said  to  be  "  in  the  same  plane  "  as  the  corresponding  surface 
of  the  other  table.  The  same  could  be  said  with  reference 
to  any  other  surfaces  answering  the  same  test.  Any  number 
of  flat  surfaces  are  said  to  be  in,  or  to  "lie  in,"  the  same 
plane  with  one  another,  and  the  same  is  true  of  any  lines  or 
points  used  to  define  any  surface  or  position  in  that  plane. 

The  planes  in  the  foregoing  illustration  of  the  floor  and 
tables  are  horizontal  planes,  but  may  be  imagined  in  any 
position,  vertical  or  inclined,  needed  for  the  projections  of 
a  drawing. 

45.  How  the    Position  of  a    Plaue    Is   Determined. 

Since  any  two  points  determine  the  position  of  a  line,  so  any 
three  points  not  in  the  same  straight  line  determine  the 
position  of  a  plane.  To  illustrate:  Take  a  square  piece  of 
'cardboard,  thick  enough  to  remain  flat,  and  push  pins  of 
equal  length  through  each  of  the  four  corners  so  that  they 
will  resemble  the  legs  of  a  chair.  The  object  will  stand 
firmly  when  placed  on  a  level  surface  with  the  points  of  the 
pins  down,  for  the  reason  that  all  the  points  represented 
by  the  ends  of  the  pins  are  in  the  same  plane.      If  one  of  the 


38  PRACTICAL    PROJECTION  §  15 

pins  is  withdrawn  and  a  shorter  one  inserted  in  its  place, 
the  cardboard  will  not  be  stable  when  placed  as  before,  and 
can  be  "  rocked."  for  the  point  at  the  extremity  of  the  short 
pin  is  not  in  the  same  plane  with  the  other  three.  Two 
planes  are  thus  defined — one  determined  by  points  at  the 
extremities  of  the  three  long  pins  and  the  other  by  points  at 
the  ends  of  the  short  and  the  two  adjacent  pins.  Both  of 
these  planes  may  be  imagined  as  extended  indefinitely,  one 
plane  being  inclined  to  and  intersecting  the  other. 

Again,  a  fiat  sheet  of  metal  may  be  supposed  to  represent 
a  plane  surface.  All  points  that  may  be  located  on  this 
sheet  are  in  the  same  plane ;  but  if  a  sheet  that  is  "  buckled  " 
is  chosen,  it  is  possible  to  locate  some  points  on  the  surface 
of  that  sheet  higher  or  lower  than  others,  and  the  points 
would  then  be  in  different  planes.  The  connection  between 
the  plane  and  the  plane  surface,  then,  is  such  that,  to  be 
defined  as  a  plane  surface,  every  point  on  that  surface  must 
be  in  the  same  plane. 

46.  In  drawing  different  views  for  the  illustration  of 
the  plane  surface,  we  shall  first  use  the  octagon,  requiring  the 
projection  of  eight  points  and  the  intermediate  lines.  The 
use  of  the  word  "imaginary  "  in  connection  with  the  state- 
ment of  the  problem  will  henceforth  be  discontinued,  since 
it  has  been  clearly  shown  that  all  surfaces,  as  well  as  other 
geometrical  elements,  depend  for  their  existence  on  the 
imaginative  feature  referred  to  in  previous  articles.  It 
will  be  understood,  therefore,  when  any  geometrical  element 
is  mentioned,  that  the  practical  feature  is  to  be  employed — 
the  imaginative,  of  course,  being  implied. 


PROBLEM   5 

47.  To  project  three  vie^vs  of  an  octagonal  sur- 
face, reiDresenting  it  in  a  liorlzontal  position. 

The  three  views  consist  of  a  plan,  front,  and  side  eleva- 
tion. A  perspective  view  of  the  surface  in  the  required 
position  is  shown  in  Fig.  15. 


§15 


PRACTICAL    PROJECTION 


39 


ExPLAXATiON. — All  lines  used  to  define  this  surface  in  the 
plan  are  at  right  angles  to  the  vertical  lines  of  sight ;  and 
since  the  line*^  will  thus  be  drawn  in  their  full  length  in  that 
view,  the  surface  will  there  be  shown  in  its  full  dimensions. 


Fig. 15 

This  principle  also  applies  to  any  view  of  a  plane  surface  in 
Avhich  all  its  lines  are  at  right  angles  to  the  lines  of  sight. 
The  plan  of  the  surface,  then,  will  be  a  true  octagon,  and 
may  be  drawn  on  the  plate  with  lines  tangent  to  a  circle 
1^  inches  in  diameter,  using  the  T  square  and  45°  triangle 
for  that  purpose. 

CoxsTRUCTiox. — Draw  the  base  line  for  the  front  elevation 
3  inches  above  the  lower  border  of  the  drawing,  and  draw 
the  vertical  base  line  (for  the 
side  elevation)  2f  inches  from 
the  left-hand  border.  De- 
scribe the  circle  previously 
mentioned  in  such  a  position 
that  the  nearest  edges  of  the 
octagon  will  be  ^  inch  from 
each  base  line ;  the  figure  may 
then  be  completed  in  the  plan. 
In  this  and  the  remaining 
problems  to  be  drawn  on  this 
plate,  the  right  views  are  to  be 
drawn  ^  inch  from  the  base 
line  in  all  cases.  In  both  eleva- 
tions,   the    lines   of  sight  in   crossing  the  surface  pass  also 


40  PRACTICAL    PROJECTION  §  15 

through  the  points  in  that  portion  of  the  surface  farthest 
from  the  observer;  and  as  the  eye  of  the  observer  travels 
from  points  opposite  to  M  and  X,  Fig.  IG,  in  tracing  the 
front  elevation,  the  foot  of  every  line  of  sight  would  be 
projected  on  a  single  line  on  V  P.  The  elevation  x>i  the 
surface,  therefore,  is  represented  on  the  drawing  by  the 
single  straight  line  Jll  JV,  Fig.  16.  Project  the  front  and 
the  side  elevations  in  their  proper  places,  completing  the 
problem. 

Note. — The  sirKgle  line  that  constitutes  each  elevation  of  this  prob- 
lem represents  the  eight  lines  of  the  octagonal  surface  shown  on  the 
plan.  This  is  shown  in  Fig.  Ifi,  which  is  a  copy  of  the  plan  and  front 
elevation  on  the  plate,  lettered  for  convenience  of  reference.  Two  of 
the  lines  in  the  plan,  A  /)  and  /^  J^,  Fig.  16,  are  shown  in  their  full 
length  by  that  portion  of  the  line  J/ A' included  between  the  points  P 
and  (2  ;  since  P' E  is  directly  in  line  with  A  B  in  the  elevation,  as 
already  e.xplained,  it  is  shown  by  the  same  line  P  O  used  to  define  A  B. 
The  line  HA  is  shown  foreshortened  at  MP;  and  a.?>  G  F  i?,  directly 
behind  HA,  MP  represents  (7 /-"also;  Q  A'  bears  the  same  relation 
to  /j'Cand  ED.  The  line  G  H  is  represented  in  the  elev£rtion  by  the 
point  J/;  A\  in  like  manner,  represents  DC.  Thus,  the  line  M N 
represents  a  certain  view  of  the  eight  lines  A  B,B  C,  etc.  to  HA,  and 
also  a  view  of  the  surface  defined  by  those  lines. 

The  above  is  very  important  and  should  be  carefully  read,  as  it 
shows  the  application  of  principles  to  Problem  5. 


PROBLEM     6 

48,  To  project  tlie  vie^vs  of  an  octagonal  surface 
that   is  in  a  riglitly  inclined  position. 

Note. — No  perspective  figure  is  shown  for  this  problem,  and  the 
projections  will  be  made  on  the  plate  from  the  following  directions. 
The  same  figure  is  used  for  this  problem  as  for  Problem  5.  These 
drawings  are  really  a  continuation  of  that  problem;  and  since  a  full 
view  of  the  surface  is  shown  in  the  plan  of  Problem  5,  projectors  will 
be  drawn  from  that  view,  in  order  to  define  the  plan  of  this  problem. 

Construction. — Draw  the  horizontal  base  line  in  the  next 
space  on  the  drawing  and  at  the  same  distance  from  the 
lower  edge  as  the  corresponding  base  line  was  drawn  in  the 
space  for  Problem  5;  draw  the  vertical  base  line  If  inches 
from  the  left  side  of  the  space. 

The  front  elevation  of  this  problem  will  first  be  drawn. 
Draw  a  line  inclined  to  the  horizontal  base  line  at  an  angle 
of  G0°  and  equal  in  length  to  the  line  shown  in  the  front 


§  15  PRACTICAL    PROJECTION  41 

elevation  of  Problem  o;  the  lower  end  of  this  line  should 
be  ^  inch  above  the  base  litie,  as  previously  explained.  It 
should  be  drawn  in  such  a  position  on  the  plate  that  vertical 
projectors  from  the  ends  of  the  line  will  pass  through  the 
central  portion  of  the  horizontal  base  line.  Mark  the  posi- 
tion of  the  points  indicated  by  the  projectors,  as  at  Jlf,  P,  Q, 
and  y,  Fig.  10,  and  from  these  points  draw  four  vertical 
projectors  to  the  plan.  Intersect  these  with  horizontal  pro- 
jectors drawn  from  the  plan  in  Problem  5;  draw  the  con- 
necting lines  between  corresponding  points  thus  projected; 
this  produces  a  figure  that  is  the  plan  of  the  octagonal  sur- 
face in  the  rightly  inclined  position  indicated  by  the  front 
elevation. 

Project  the  side  elevation  by  the  use  of  secondary  pro- 
jectors, as  previously  explained.  Reference  to  the  copy  of 
this  plate  will  be  of  assistance  to  the  student  during  the  pro- 
jection of  these  views.  The  completed  drawings  are  there 
shown  and  the  method  of  projecting  between  different 
views  is  indicated  by  projectors  partially  extended  towards 
the  left  of  the  plan  of  this  problem. 

49.  Basis  of  Projection. — The  plan  and  side  eleva- 
tion of  this  surface  are  foreshortened  views.  There  are, 
however,  two  lines  in  each  view  shown  in  their  true  length; 
this  may  be  proved  by  a  comparison  of  the  figures  with  the 
plan  of  Problem  o;  the  other  lines  in  each  case  are  fore- 
shortened. 

It  does  not  necessarily  follow,  however,  that  any  of  the 
lines  in  a  rightly  inclined  view  are  shown  in  their  true  length. 
Had  the  angle  of  inclination  been  along  the  line  A  E,  Fig.  IG, 
every  line  would  have  been  foreshortened;  again,  in  the 
case  of  surfaces  having  curved  or  irregular  outlines,  the  least 
angle  of  inclination  in  any  direction  would  preclude  the 
possibility  of  representing,  in  a  foreshortened  view,  any 
portion  of  the  outline  in  its  true  length. 

It  will  thus  be  seen  that  the  point  is  the  only  geometrical 
element  not  sul)ject  to  change,  or  variation,  in  any  view. 
It  may  therefore  be  relied  on  as  a  basis  of  projection.      The 


42  PRACTICAL    PROJECTION  §  15 

outline  of  any  surface  in  the  different  views  is  determined 
by  first  fixing  the  location  of  points  at  the  extremities  of 
the  boundary  lines  of  such  surface,  afterwards  drawing  the 
connecting  lines,  as  in  this  problem. 


PROBLEM    7 

50.  To  pi'Qject  a  full  A-ie^v  of  a  surface  from  a 
given  plan  and  elevation  slio'wing;  that  surface  in  a 
rightly  inclined  position. 

A  full  view  of  any  surface  may  be  projected  by  assuming 
a  view  to  be  taken  at  right  angles  to  a  line  in  which  the 
entire  surface  is  represented,  for  in  such  a  view  the  lines  of 
sight  are  at  right  angles  to  the  outlines  of  the  surface. 

CoNSTRUCTiox.— In  the  next  adjoining  space  to  the  right 
on  the  plate,  copy  the  plan  and  the  front  elevation  of  Prob- 
lem G,  placing  the  projections  so  that  they  will  occupy  the 
same  relative  position  in  the  space.  To  obtain  a  full  view 
of  the  surface  in  this  problem,  a  view  must  now  be  assumed 
at  right  angles  to  the  line  in  the  elevation ;  in  other  words, 
the  elevation  must  be  considered  as  a  plan  and  a  new 
front  elevation  projected  therefrom.  The  plan  copied  from 
Problem  G  is  used  as  a  base  plan,  secondary  projectors  being 
drawn  from  thence  in  the  manner  shown  on  the  plate  and 
described  in  the  following  article.  The  projectors  in  this 
case  are  drawn  by  the  arc  method,  sometimes  more  con- 
veniently employed  than  the  angular  method  previously 
described. 

51.  Arc  Method  of  Drawing  Secondary  Project- 
ors.— Draw  a  base  line,  for  the  projection  of  the  full 
view,  from  the  intersection  of  the  base  lines  previously 
drawn  and  parallel  to  the  line  that  represents  the  surface  of 
the  octagon,  as  A  B  (see  plate).  At  the  point  of  intersec- 
tion of  the  base  lines  {B)  erect  a  perpendicular  to  the 
oblique  base  line  A  B,  as  B  C,  producing  it  indefinitely 
towards  the  right.      The  positions  of  all  points  in  the  plan 


§  15  PRACTICAL    PROJECTION  43 

are  now  to  be  located  on  this  line  in  the  same  relative  posi- 
tion as  they  would  occupy  if  projected  horizontally  to  the 
vertical  base  line  in  the  drawing.  The  points  are  accord- 
ingly projected  horizontally  to  the  vertical  base  line  B  I); 
thence,  by  using  the  compasses  and  describing  arcs  from  a 
center  B,  located  at  the  intersection  of  the  base  lines,  they 
are  projected  to  the  line  B  C.  The  projectors  are  then  con- 
tinued beyond  B  C,  but  parallel  to  A  B;  they  are  there 
intersected  by  primary  projectors  drawn  from  corresponding 
points  in  the  elevation,  as  shown.  Locate  the  various  posi- 
tions of  the  corresponding  points  at  the  intersections  of  these 
projectors  and  produce  the  full  view  of  the  octagon  by 
drawing  the  connecting  lines. 

It  will  thus  be  seen  that  the  drawing  of  secondary  pro- 
jectors by  this  method  involves  Jirst,  projection  to  the 
nearest  base  line ;  second^  the  describing  of  arcs  from  the 
center  shown;  and  third,  the  continuation  of  the  projectors 
parallel  to  the  base  line  of  the  desired  view.  If  the  draw- 
ing has  been  carefully  made,  it  will  be  found  that  the  sur- 
face thus  defined  is  an  exact  counterpart  of  the  plan  in 
Problem  5,  and  that  the  full  view  projected  in  this  problem 
is  in  the  same  relation  to  the  elevation  as  the  plan  in  Prob- 
lem 5  is  to  the  elevation  of  that  problem. 

53.  Tie>vs  Xecessary  for  the  Projection  of  the 
Full  Vie\v. — When  it  is  desired  to  project  a  full  view  of 
any  surface  that  is  represented  in  a  drawing  in  an  inclined 
position,  it  is  necessary  to  have  one  view  that  will  show  all 
the  points  of  that  surface  as  contained  in  one  line.  A  pro- 
jection must  also  be  drawn  at  right  angles  to  that  view,  in 
order  that  such  dimensions  of  the  surface  as  are  at  right 
angles  to  those  in  the  first  view  may  be  shown  in  their  true 
length. 

53.  Full  A'ie^vs  Sometimes  Obtained  Without  Pro- 
jection 3Ietlio(ls. — A  comparison  of  the  views  in  the 
projections  of  the  last  problem  will  prove  that  the  vertical 
primary  projectors  included  within  the  surface  of  the  octa- 
gon shown  in  the  plan  are  of  the  same  length  as  the  secondary 


44  PRACTICAL    PROJECTION  §  15 

projectors  in  the  view  last  projected.  This  knowledge 
may  be  used  to  some  advantage  in  producing  a  full  view 
without  using  all  the  projectors  employed  in  this  problem. 

Thus,  draw  a  horizontal  center  line  through  the  plan,  as 
E  F  (see  drawing  on  the  plate  for  Problem  7)  and  draw 
primary  projectors  from  the  elevation  to  the  full  view  in 
the  regular  way;  at  a  convenient  distance  draw  a  line  at 
right  angles  to,  and  crossing,  these  projectors,  as  G  H. 
This  line  will  be  the  center  line  for  the  full  view,  the  points 
of  which  may  then  be  located  with  the  dividers  in  the  fol- 
lowing manner:  Set  the  dividers  to  the  length  ac  xn  the 
plan  and  set  off  a  corresponding  distance  at  a  e'  in  the  full 
view;  in  like  manner  make  b'  f  equal  to  b  f,  etc.,  as  shown 
on  the  plate.  Complete  the  outline  of  the  full  view,  then, 
by  drawing  connecting  lines  as  heretofore. 

This  method  is  generally  followed  in  pattern  drafting, 
since  it  requires  less  time  than  to  draw  full  projections  as  in 
the  construction  of  the  problem,  and  there  is  less  liability 
of  error. 

54.  If  the  student  that  does  not  clearly  understand  the 
principles  by  which  these  projections  are  made  will  cut  a 
piece  of  cardboard  to  the  same  size  and  shape  as  the  plan 
of  Problem  o  and  hold  it  in  such  positions  that  the  foot 
of  the  lines  of  sight  falls  on  the  points  designated  on  the 
drawings  for  the  different  views,  he  will  at  once  see  the 
correct  position  of  the  surface  as  represented  in  each  view. 

oo.  Surfaces  Bounded,  by  Curved  Lines. — Surfaces 
that  are  defined  by  curved  lines  do  not  present  any  points 
from  which  to  make  projections.  In  making  such  pro- 
jections, the  same  principles  are  employed,  however;  but  it 
is  first  necessary  to  establish  a  number  of  points  at  various 
positions  on  the  curved  lines.  The  points  thus  established 
are  then  projected  in  the  same  way  as  in  the  foregoing 
problems.  "When,  for  purposes  of  projection,  points  are 
located  on  the  outline  of  a  curved  surface  in  any  view,  it 
should  be  observed  that  they  are  so  placed  that,  when  the 


§  15  PRACTICAL    PROJECTION 


45 


points  thus  located  are  projected  to  a  line  that  represents 
an  edge  view  of  that  surface,  each  end  of  that  line  is  defined 
by  the  projection  of  a  point. 


PUOHLEM   8 

56.  To  project  views  of  a  plane  surface  defined  by 
a  curved  line,  the  snrfiice  being  in  a  rightly  inclined 
position;  also,  to  project  a  point  located  on  that 
surface. 

Note.— Before  making  the  projections  of  this  problem,  the  lines  to 
remain  on  the  drawing  for  Problem  5  should  be  inked  in,  and  to  avoid 
confusion,  all  other  lines  not  to  be  inked  in  on  that  figure  should  be 
erased;  the  circle  drawn  for  Problem  5  may  then  be  redrawn  for 
this  problem. 

Construction.— The  surface  for  the  projections  of  this 
problem  is  that  of  the  circle  to  which  the  sides  of  the  octa- 
gon in  Problem  5  are  tangent.  After  describing  the  circle, 
the  next  step  is  to  locate  points  on  its  circumference.  Do 
this  by  first  drawing  a  vertical  and  a  horizontal  diameter, 
and  then  drawing,  with  the  45°  triangle,  two  other  diam- 
eters at  right  angles  to  each  other,  thus  locating  eight  points 
at  equal  distances  on  the  circumference.  Those  points 
indicated  by  the  horizontal  diameter  will,  when  projected 
to  the  elevation,  define  the  ends  of  the  line  in  that  view. 

Also,  locate  a  point  at  the  center  of  the  circle.  The  plan 
and  elevation  of  the  surface  thus  projected  is  shown  in 
Fig.  17,  the  points  being  denoted  by  numerals.  Project 
these  points  to  the  elevation  of  Problem  5,  using  lines 
easily  erased,  since  they  are  not  to  appear  in  that  problem 
when  the  plate  is  finished.  In  the  last  space  on  the 
plate  draw  horizontal  and  vertical  base  lines  in  the  same 
corresponding  position  as  in  the  space  for  Problem  G.  The 
line  that  represents  the  front  elevation  of  the  circular  plane 
surface  is  then  drawn  in  the  same  position  as  in  the  ele- 
vation in  Problem  G;  with  the  dividers,  locate  points 
thereon  in  the  same  position  as  the  points  projected 
from   the  circle   to   the    front   elevation    in    Problem    5,    as 


M.  E.     V.—J5 


46 


PRACTICAL    PROTECTION 


§15 


1     3 


Fig. it 


shown  in  Fig.  IT.  Project  these  points  vertically  to  the 
plan  and  intersect  these  projectors  with  horizontal  pro- 
jectors drawn  from  the 
circle  in  Problem  5. 

When  projecting  points 
across  a  drawing — as  from 
the  space  occupied  by  Prob- 
lem 5  to  the  drawing  for 
this  problem  —  it  is  not 
necessarj-  to  draw  lines  the 
entire  distance.  By  care- 
fully placing  the  edge  of 
the  T  square  on  each  point 
in  turn,  corresponding  lines 
ma)'  be  drawn  across  the 
plan  in  this  problem.  This 
saves  erasing  unnecessary  lines,  but  care  must  be  taken, 
when  making  projections  in  this  way,  to  observe  that  points 
thus  located  are  at  the  intersections  of  projectors  drawn 
from  points  in  corresponding  positions  in  each  of  the  views. 
Find  the  location  of  each  point  thus  projected  and  through 
these  points  trace  the  curve  that  represents  the  foreshort- 
ened view  of  the  circle,  using  the  irregular  curve  for  this 
purpose.  Thus  a  plan  and  a  front  elevation  of  the  circular 
plane  surface  is  drawn,  in  which  the  surface  is  represented 
in  a  rightly  inclined  position. 

Project  the  side  elevation  by  the  angular  method  of  sec- 
ondary projectors,  as  previously  explained,  and  designate 
the  point  in  both  views  by  a  small  dot  at  the  center  of  the 
surface.  Finally  project  the  full  view  by  the  arc  method, 
as  in  Problem  T.  In  this  problem  a  good  test  of  accuracy 
is  afforded  if,  after  the  nin'e  points  have  been  projected  to 
the  full  view,  a  circle  with  a  radius  of  f  inch,  described  from 
the  central  point,  passes    through    the  other    eight   points. 


57.  Importance  of  Acciiiticy. — Next  to  a  knowledge 
of  the  principles  of  projection,  neatness  and  accuracy  are 
the   prime   requisites   in  a   drawing.      The   student   should 


PRDJEC 


:^^V.l   \  \\\    1  I       I      /ft 

;  /  / //  :vV'\  \V5/^-t 


I  \  \  \  1  \  \ 


/ 
/// 


\  J^^ 


X 


PRC 


PROBLEM  9. 


PROBLEM  12. 
CaseL 


PROBLEM  12. 
Case  2. 


JUNE  85,/ 893. 


Copyright,  l^•99,  by  THE  Col 
All  right 


noNS-n. 


-^=^T\ 


^ 


<\A  \ 


\      ^ 


/ 


/ 


U./^'/^^^yyc^ 


^ 


^ 


.^ 


EM  10. 


Pf^OBLEM  II. 


PROBLEM  12. 
Case  3. 


PROBLEM  IS. 
Case  4. 


EK^•  ICnginf.er  Company. 
served. 


JOE/N  3MLTHj  CL  A55  N2  ^529, 


§  15  PRACTICAL    PROJECTION  47 

carefully  observe  that,  when  the  points  determined  by  the 
intersection  of  lines  are  used  as  centers  for  arcs  or  circles, 
the  needle  point  of  the  compasses  should  be  placed  exactly 
on  that  position;  again,  drawing  three  or  more  lines  that 
shall  intersect  at  the  same  point  is  very  commonly  required 
in  projection  drawing  and  in  pattern  drafting;  this  is  not 
an  easy  thing  to  do  accurately  unless  carefully  practiced  by 
the  student.  It  is  needless  to  state  that  unless  the  work  is 
accurately  done  it  is  of  no  value. 

When  putting  in  the  figures  for  the  dimensions  on  draw- 
ings, care  should  be  observed  that  they  are  placed  on  those 
views  in  which  the  lines  and  surfaces  are  shown  in  their 
true  length.  Do  not  designate  a  foreshortened  view  of  a 
line  or  surface  by  a  dimension  figure,  when  another  view  is 
given  in  which  the  true  length  is  shown.  Again,  do  not 
repeat  the  same  dimension  on  different  views  of  the  same 
drawing;  thus,  in  Problem  2,  Case  I,  the  length  of  the  line 
is  given  as  2  inches  in  the  plan,  and  it  is  obviously  unneces- 
sary to  give  the  same  dimension  in  the  front  elevation. 

The  student  may  ink  in  all  the  problems  on  the  plates, 
but  the  letters  used  to  describe  the  different  positions  and 
lines  are  not  placed  on  the  drawing.  The  date,  name,  and 
class  letter  and  number  are  inscribed  as  in  the  plates  of 
Geometrical  Drazvinsr. 


DRAAVING    PIRATE,    TITTLE:    PROJECTIONS    II 

58.  The  problems  for  this  and  the  succeeding  plates 
should  be  practiced  on  other  paper  and  then  copied  on  the 
drawing  that  is  to  be  sent  in  for  correction.  The  student 
can  thus  judge  better  as  to  the  relative  position  the  figures 
sljould  occupy  and  the  completed  plates  will  present  a  neat 
appearance.  In  making  the  projections  on  this  and  the 
succeeding  plates,  the  views  may  be  assumed  to  be  \  inch 
from  their  respective  base  lines,  as  this  will  enable  the  pro- 
jections to  be  kept  in  closer  proximity.  The  base  lines  are 
not  to  be  inked  in  on  this  or  the  following  plates.  Divide 
this    plate    by    a    central    horizontal    line;    the   part   of   the 


48 


PRACTICAL    PROJECTION 


15 


drawing  above  this  line  is  divided  into  three,  and  the  part 
below  the  line  into  four,  equal  spaces. 


PROBLEM  9 

59.  To  project  a  side  elevation  and  a  full  view 
of  a  rightly  inclined  plane  surface  defined  by  an 
irregular  outline. 

This  is  a  problem  in  which  the  student  has  an  opportu- 
nity to   use,   in   a   practical  way,   the 
^Elevation  knowledge     of     projection     thus    far 

gained.      The  surface  to  be  projected 
is  shown  in  a  rightly  inclined  position 

— -i-j-j-^i — f    M  ri ^^^    ^^S-   18,  which   is   a   foreshortened 

1 1  jdi^l  f  J<^  I  i  .  t      view  of  the  surface.      This  figure  is  to 
jyf'fv^r^j'/^^  be  copied,  in  the  size  indicated  by  the 

^li  i  i  I"  i**  I  I  ;  iW^t  dimension  figures,  into  such  a  position 
in  the  upper  left-hand  space  on  the 
drawing  that  the  projections  when 
completed  will  occupy  about  the  cen- 
ter of  the  space. 

Construction. — First  draw  the 
horizontal  line  A  B  1^  inches  long  and 
bisect  it  at  o  by  the  vertical  line  C  D.  Make  o  D  \  inch 
long  and  bisect  it  at  x,  as  shown,  making  o  C  1^  inches 
long.  Set  the  compasses  at  a  radius  of  Iff  inches,  and 
with  A  and  C,  respectively,  as  centers,  describe  arcs  inter- 
secting at  e;  with  the  same  radius  and  with  B  and  C  as 
centers,  describe  arcs  similarly  intersecting  at  e' .  From 
these  centers  {e  and  e')  describe  the  arcs  A  C  and  B  C, 
thus  producing  the  curved  outline  of  the  lower  portion  of 
the  plan.  Next,  divide  these  arcs,  by  spacing,  into  six 
equal  parts,  thus  locating  the  points  1,  2,  3,  4,  and  ,5; 
from  these  points  draw  vertical  lines,  as  shown  in  Fig.  18. 
Complete  the  upper  outline  of  the  surface  as  represented  in 
the  figure;  thus,  locate  a  at  the  intersection  of  the  vertical 
from  1  with  a  horizontal  from  A  ;  d,  at  the  intersection  of 


§  15  PRACTICAL    PROJECTION  49 

the  vertical  from  ;?  with  a  horizontal  from  D\  /",  in  like 
manner,  at  the  intersection  of  a  vertical  from  J  with  a  hori- 
zontal from  X. 

The  points  at  the  extremities  of  these  lines  and  those 
located  on  the  curved  outline  are  now  to  be  treated  as  in 
former  problems  and  the  projections  made  in  the  usual  way. 

Caution. — When  making  the  projections  for  this  prob- 
lem, the  student  must  observe  the  precautions  given  in 
regard  to  the  taking  of  the  same  corresponding  points  in 
each  view.  Project  the  side  elevation  first ;  it  may  be  desir- 
able for  the  student  to  ink  in  that  figure,  in  order  to  avoid 
the  confusion  arising  from  a  number  of  lines  crossing  one 
another  on  the  drawing.  Use  the  angular  method  for  the 
secondary  projectors  in  projecting  the  side  elevation  and 
the  arc  method  for  the  full  view.  Since  it  is  often  neces- 
sary, when  developing  patterns,  to  draw  several  views  over 
one  another  in  this  way,  the  student  should  accustom  him- 
self to  drawings  that  have  a  complicated  appearance  from 
this  cause,  and  should  learn  to  follow  each  set  of  projectors 
as  readily  as  though  they  were  in  separate  drawings. 
During  the  construction  of  this  projection,  it  will  be  noticed 
that  the  base  line  for  the  full  view,  in  order  to  be  drawn 
from  the  intersection  of  the  other  base  lines,  will  fall  below 
the  front  elevation  of  the  surface.  This  is  unimportant,  how- 
ever, since  its  purpose  is  the  same,  and  the  result  is  merely 
that  of  a   slight  appearance  of   crowding  on  the  drawing. 


PROBLEM    10 

60.  To  project  views  of  a  plane  surface  in  an 
obliquely  inclined  position. 

Explanation.— A  full  view  of  the  surface  to  be  projected 
in  this  problem  is  shown  in  Fig.  19,  the  dimension  figures 
giving  the  size  in  which  it  is  to  be  drawn  by  the  student. 
The  upper  portion  of  this  surface  is  defined  by  a  semicircle, 
the  lower  by  one-half  of  an  octagon.  The  purpose  in 
selecting  a  surface  of  this  outline   is  to  give    the    student 


50 


PRACTICAL    PROJECTION 


5  15 


some  practice  in  the  projection  of  both  straight  and  curved 

outline  surfaces. 

It   has   been   shown  that,   before    an  inclined   view  of  a 

surface  was  projected,  a  right  view — i.  e.,  a  right  plan 
and  elevation,  as  in  Problem  5 — has  first 
been  drawn.  These  views  alone  are  pro- 
jected in  drawings  of  simple  or  plain 
4j>  objects,  it  being  obviously  unnecessary  to 
show  any  object  in  a  working  drawing  in 
a  position  not  commonly  occupied.  But 
owing  to  the  different  shapes  of  objects, 
variously  outlined  surfaces  are  presented  in 

a  diversity  of  positions,  and  it  is  essential  that  the  student 

should   be    capable    of    projecting    any   surface    into    any 

conceivable  position  and  of  drawing  a  full  view  from  such  a 

projection. 

CoxsTRUCTiox. — The  method  of  drawing  oblique  views  of 

surfaces  is  shown  in  detail  at  (a).  (^).  and  (c).  Fig.  20,  the 


Fig.  19 


-2> 


projections  at  (c)  being  the  ones  required  for  the  plan  and 
elevation  of  this  problem.  Lay  a  separate  piece  of  paper 
over  the  drawing  of  Problem  9  on  the  plate,  and  reproduce 
thereon  the  projections  shown  at  (a)  and  (^),  Fig.  20,  in 
accordance  with  principles  already  explained.      Next,  draw 


§  15  PRACTICAL    PROJECTION  51 

the  plan  and  elevation  at  {c)  in  their  proper  places  on  the 
plate.  The  drawing  shown  at  (a)  may  be  seen  to  be  similar 
to  that  of  Problem  5  of  the  preceding  plate;  (d)  is  projected 
directly  from  (a),  in  the  same  manner  as  Problem  6,  the 
angle  of  inclination  being  60°.  The  plan  of  this  surface  in 
(/?)  is  then  copied  at  (c)  in  such  a  position  that  the  center 
line  A  B  makes  an  angle  of  60°  with  the  base  line  of  V  P. 
This  is  accomplished  by  first  drawing  the  center  line  A'  B' 
at  the  given  angle  in  (r),  noting  thereon  the  position  of  the 
points  w',  x\  and  y' \  draw  perpendiculars  through  these 
points,  and  make  zv'  a'  in  {c)  equal  to  w  a  in  (/;),  x'  D'  equal 
to  X  D,  etc.  The  outline  of  the  plan  at  {c),  therefore,  is 
exactly  the  same  as  it  is  shown  in  {b),  the  only  difference 
between  the  two  views  being  the  fact  that  the  line  A'  B'  in  {c) 
is  inclined  to  V  P,  while  in  {b)  it  is  perpendicular  to  that  plane. 

Let  us  consider  what  changes  have  here  been  represented. 
Cut  a  piece  of  cardboard  to  the  outline  and  size  shown  in 
Fig.  19  and  compare  it  with  the  different  positions  in  the 
drawings  just  made.  It  will  be  seen  that  the  cardboard 
must  be  held  in  a  horizontal  position  to  coincide  with  the 
drawing  at  {a);  to  represent  the  drawing  at  {b),  the  point  C 
must  be  raised  until  the  line  CD  is  at  the  angle  of  60° 
with  H  P.  The  plan  at  {b)  is,  therefore,  a  foreshortened 
view  of  the  surface,  although  its  elevation  may  still  be  rep- 
resented by  the  single  line  C  D' .  Now  turn  the  cardboard 
to  the  position  indicated  in  (^),  that  is,  so  that  the  line  A'  B' 
makes  an  angle  of  60°  with  V  P. 

It  will  be  seen  that  the  line  C  D  '\n  its  relation  to  H  P  is 
not  affected  by  this  change,  its  angle  with  H  P  remaining  as 
before;  therefore,  the  vertical  distances  to  be  sJioivn  in  the 
elevation  of  {c)  will  be  the  same  as  in  the  elevation  of  {b), 
and  may  be  projected  directly  to  {c)  from  {b),  as  shown  in 
Fig.  20.  Draw  horizontal  projectors  from  the  elevation 
at  (b)  to  the  elevation  in  (c),  intersecting  them,  in  the  man- 
ner shown,  by  primary  projectors  drawn  vertically  upwards 
from  the  points  in  the  plan  at  (c).  Trace  the  outline  of  the 
surface  thus  indicated  through  the  intersections  of  project- 
ors drawn  from   corresponding  points  in  each   view.     The 


52  PRACTICAL    PROJECTION  S$  15 

projections  shown  at  (r)  being  completed  on  the  plate,  the 
paper  on  which  (a)  and  (d)  were  drawn  may  now  be 
removed  and  the  side  elevation  required  for  the  problem 
projected  by  the  angular  method  previously  described. 
Three  views  are  thus  shown,  in  all  of  which  the  surface  is 
represented  as  inclined  at  an  oblique  angle  to  the  lines  of 
sight;  all  these  views,  therefore,  are  foreshortened.  Oblique 
views  may  always  be  drawn  in  this  manner;  that  is,  a  right 
view  is  first  drawn;  next,  a  rightly  inclined  view  is  pro- 
jected, the  desired  angle  being  represented  in  the  elevation. 
The  plan  thus  produced  is  then  redrawn  for  the  oblique 
view  and  its  elevation  projected  as  in  this  problem. 

61.     Position    of     Full    Views:     How    Determined. 

To  project  a  full  view  of  this  surface  it  is  first  necessary  to 
determine  whether  any  of  the  lines  or  distances  in  any  of 
the  views  are  shown  in  their  true  length,  but  without  hav- 
ing recourse  to  the  projections  made  on  the  separate  paper, 
since  projection  methods  are  to  be  used.  This  may  be  done 
by  comparing  the  relative  position  of  any  two  points  in  the 
outline  of  the  surface,  as  located  in  the  plan  and  elevation. 
If  it  is  found  that  a  line  drawn  between  any  two  of  these 
points  in  the  elevation  will  be  parallel  to  the  base  line  (and 
therefore  at  right  angles  to  the  vertical  lines  of  sight),  that 
line  will  be  shown,  of  course,  in  its  true  length  on  the  plan. 
Any  other  lines  parallel  to  it  will  also  be  shown  in  their 
true  length.  It  is  found  on  examination  that  points  in  the 
elevation  corresponding  to  the  positions  represented  by 
A  and  B,  Fig.  19,  are  located  on  the  same  horizontal  pro- 
jector; therefore,  a  line  drawn  between  these  points  as  they 
are  located  on  the  plan  will  be  represented  in  its  true  length 
in  that  view;  and  a  view  projected  from  these  points  in  the 
plan  by  primary  projectors  drawn  at  right  angles  to  this 
line,  intersected  by  secondary  projectors  from  the  front 
elevation  (by  a  modification  of  the  method  used  in  Prob- 
lem. 7),  will  be  a  full  view  of  the  surface. 

Draw  the  oblique  base  line  in  the  proper  position,  i.  e., 
parallel  to  that  line  shown  in  full  length  in  the  plan  [A  B  on 


§  15  PRACTICAL    PROJECTION  53 

the  plate),  as  above  explained,  and  at  such  distance  away 
from  the  plan  as  directed  in  the  instructions  for  drawing 
this  plate,  producing  the  line  indefinitely  towards  the  upper 
portion  of  the  drawing,  as  shown  on  the  plate  at  E  F.  In 
this  case,  the  line  thus  drawn  defines  the  inclination  of  the 
surface,  since  the  angle  is  the  same  in  both  plan  and  eleva- 
tion, viz.,  60°.  The  full  view  is  projected  as  follows:  Draw 
the  line  G  H  at  right  angles  to  the  base  line  EF  and 
from  the  intersection  oi  E  F  with  the  horizontal  base  line. 
By  the  arc  method,  draw  secondary  projectors  from  the 
elevation,  as  shown;  intersect  these  projectors  by  primary 
projectors  drawn  from  the  plan  at  right  angles  to  E  F,  thus 
producing  the  full  view  of  the  surface.  It  will  be  found 
that  this  full  view  is  an  exact  counterpart  of  the  surface 
shown  in  Fig.  19,  and  should  correspond  to  the  preliminary 
drawing  in  the  plan  at  (a)  on  the  separate  paper.  Note 
that  the  portion  of  EF  included  between  /  and  q  cor- 
responds in  length  to  that  of  the  rightly  inclined  front 
elevation  at  (-z^),  Fig.  20.  This  may  be  seen  by  comparing 
that  portion  of  the  line  with  the  view  on  the  preliminary 
drawing.* 

The  student  should  now  be  able  to  recognize  any  view  of 
a  surface  in  any  position;  that  is,  he  should  be  able  to  tell 
whether  a  view  represents  such  a  surface  in  a  right  position, 
a  rightly  inclined  position,  or  an  obliquely  inclined  position; 
and  by  the  application  of  the  principles  illustrated  in  the 
foregoing  problems  and  the  exercise  of  a  little  judgment, 
he  should  be  able  to  project  any  surface  into  any  desired 
position.  Or,  being  given  a  surface  in  a  position  indicated 
by  a  properly  projected  plan  and  elevation,  he  should  be 
able  to  produce  the  full  view  and  designate  the  angle  of 
inclination. 

The  following  problem  will  serve  as  a  test  of  his  progress. 
The  principles  involved  have  already  been  presented  and 
the  method  of  application  will  be  readily  understood.      The 


*  It  should  be  noted  that  this  is  the  case  only  when  the  angle  of 
inclination  is  the  same  in  both  views. 


54  PRACTICAL    PROJECTION  §  15 

angle  of  inclination  in  the  plan  is  not  the  same  as  is  shown 
in  the  elevation ;  both  angles  are  to  be  determined  by  pro- 
jection methods. 


PROBLEM    11 

62.     To    project    the    fnll    view    of    an    irregularly 
outlined  surface  obliquely  Inclined. 

The  projections  of  this  problem  are  to  occupy  the  upper 

right-hand  space  of  the  drawing  plate.      The  plan  and  front 

elevation  are  shown  in  Fig.  -21  and  are  reproduced  full  size 

on  the  sheet  opposite  this  page.*     The  outline  represented 

is  frequently  used  as  a  "stay,"  or  profile,  to  which  mold- 

— -^  ings  are  formed  in  cornice  work.    Since 

/^      \,^  the  view  shown  is  known  to  be  obliquely 

5A  gj^gr^izcvN^         inclined,  its  dimensions  are  foreshort- 

__y     ened,  and  their  true  lengths  are  to  be 

^  found    by  projection  methods,    as   fol- 

\  lows: 


3f — 


I     /  \j  CoxsTRUCTiON. — The  student  should 

I  /  /  detach  the  sheet  opposite  this  page  and 

i  (  y  paste  the    plan  and  elevation  in   such 

/      X  a  position  in  the  third  space  on  the  plate 

\y  that  the    base   line  JI X,  Fig.  •21,  will 

F'<^   -'  be  2+  inches  below  the  top  border  line 

and  exactly  horizontal.      Locate  a  number  of  points  on  the 

curved  outline  in  the  elevation,    by  equally   spacing    that 

portion  of  the  figure  with  the  dividers,  as  shown  at  i.  J,  3. 

and  4  in  the  drawing  for  this  problem  on  the  plate,   and 

project  these  points  to  the  plan.     To  ascertain  the  angle  of 

inclination  in  the  plan,  draw  a  horizontal  line  through  the 

widest   portion  of  the   figure  in  the  elevation,   locating,   if 

fKSSsible.  one  end  of  the  line  at  an  angle  of  the  surface — as 

the  line  A  B.     Project   this  line  to  the  plan  at  A'  B' .   as 

explained  in  Art.  61 ;  parallel  to  this  line  erect  the  oblique 

base  line  C  D. 


*  This  sheet  is  not  inserted  in  the  bound  volume  containing  this 
Paper. 


§  15  PRACTICAL    PROJECTION  55 

The  angle  formed  by  these  lines  {A'  B'  and  C  D)  with  the 
horizontal  base  line  is  the  angle  of  inclination  of  the  surface 
to  V  P,  or  that  angle  shown  in  the  plan.  The  angle  of  incli- 
nation to  H  P,  or  that  shown  in  the  elevation,  is  most  easily- 
found  by  constructing  a  right-angled  triangle  whose  base 
and  altitude  are  equal  to  certain  distances  found  in  the  plan 
and  elevation;  that  is,  the  base  is  equal  to  the  extreme  width 
of  the  figure  in  the  plan,  taken  at  right  angles  to  the  line  of 
inclination  in  that  view  (shown  by  the  dimension  .V).  The 
altitude  is  the  vertical  height  shown  in  the  elevation  at  AI. 
Construct  this  right-angled  triangle  on  the  horizontal  base 
line  extended,  as  shown  at  N'  M',  and  locate  one  end  of  the 
base  at  D,  the  intersection  of  the  base  lines.  Extend  the 
hypotenuse  indefinitely  towards  the  right  of  the  drawing. 
Next,  intersect  primary  projectors  drawn  from  the  plan  to  the 
oblique  view  by  secondary  projectors  drawn  from  the  eleva- 
tion by  the  arc  method,  as  shown  on  the  plate.  The  full 
view  of  the  surface  is  then  traced  through  the  intersections 
of  these  projectors,  completing  the  problem. 

63.  Projection  of  Solids. — We  now  come  to  the  pro- 
jection of  solids,  which,  as  before  noted,  are  merely  various 
combinations  of  surfaces.  Projections  of  surfaces  in  a 
variety  of  positions  having  already  been  made,  we  shall 
encounter  no  new  principles  in  the  projection  of  solids,  the 
surfaces  of  which  are  projected  in  the  same  manner  as  has 
been  shown  in  preceding  problems. 

Since  lines  intersect  in  a  point,  so  surfaces  intersect  in  a 
line,  and  in  drawing  projections  of  solids  it  is  necessary  only 
to  find  the  true  projections  in  any  view,  or  set  of  views,  of 
those  lines  that  represent  the  correct  intersections  of  the 
adjacent  surfaces;  this  is  a  comparatively  easy  thing  for  the 
student  to  do,  if  he  will  use  proper  care  and  diligence  in 
the  application  of  the  principles  of  the  preceding  problems. 

64.  Projection  of  the  Cnbe  Illustrated. — Every  solid 
consists  of  a  number  of  surfaces,  each  of  which  is  differ- 
ently shown  when  the  solid  is  projected  to  the  various  views. 


56 


PRACTICAL    PROJECTION 


§15 


This  is  due  to  the  fact  that  the  observer  is  assumed  to  occupy 
a  different  position  in  each  view  of  the  solid  thus  projected. 
In  certain  positions  some  soHds  show  one  or  more  of  their 
surfaces  directly  behind  another  surface  of  the  same  size  and 
shape.  This  would  be  the  case  if  the  projections  of  the 
cube,  referred  to  in  Art.  18,  were  drawn  as  shown  in  Fig.  22. 
When  the  cube  is  in  a  right  position — that  is,  with  two  sur- 
faces horizontal,  and  that  surface  nearest  the  observer  in  a 
side  view  in  such  position  that  the  lines  defining  that  surface 
will  be  at  right  angles  to  the  lines  of  sight,  as  indicated  in 
Fig.  22 — it  is  evident  that  the  surface  parallel  to  and  behind 


Fig.  22 

the  front  surface  will  be  projected  by  the  same  lines  of  sight 
as  the  front  surface.  Therefore,  in  such  a  case,  a  projection 
of  the  front  surface  of  the  cube  is  equivalent  to  a  projection 
of  the  entire  cube.  Each  projection  of  the  cube  in  the  plan, 
front,  and  side  elevations  is  a  square,  the  sides  of  which  are 
1  inch  long,  while  the  views  are  arranged,  as  shown  in  Fig.  23, 
in  such  a  way  as  to  appear  related  to  one  another. 


65.  In  "reading"  the  projections  shown  in  Fig.  23, 
we  merely  compare  the  surfaces  of  the  cube  as  they 
are  shown  in  the  different  projections.  Thus,  the  sur- 
face A  B  C  D,  Fig.  22,  is  represented  in  the  plan  of  Fig.  23 


15 


PRACTICAL    PROJECTION 


57 


Front  Llevaiion 


IjT 


Siile  Elevation 


by  the  line  .4  />,  antl  in  the  front  elevation  by  the  line  B  C, 
while  a  full  view  of  that  surface  is  shown  in  the  side 
elevation ;  so,  in  like  manner,  the  position  of  each  surface 
of  the  cube  may  be  determined.  Note  that,  in  each  full 
view,  two  surfaces 
are  represented ;  thus, 
in  the  front  elevation 
of  Fig.  '>2,  the  sur- 
faces />  F  G  C  and 
A  E  H D  are  projected 
on  V  P  as  one  surface 
at  PQRS.  It  is  thus 
shown  that  surfaces  in 
their  relation  to  one 
another,  when  combined 
in  one  view,  as  in  the 
projection  of  a  solid, 
partake     of     the     same     •"  y         ^  c  a 

principle   that  has  been  '°'  "" 

shown  in  its  application  to  points  and  lines,  viz.,  surfaces 
ivhose  outlines  are  contained  in  the  same  lines  of  sight  in 
any  view  are  projected  in  that  viezv  as  one  sjirfacc. 

The  difference  in  position  of  the  several  views  due  to  the 
use  of  first-  and  third-angle  projection  may  easily  be  illus- 
trated in  connection  with  Fig.  23.  Here  the  cube  is  repre- 
sented as  having  been  projected  in  the  first  angle.  If  the 
student  will  now  turn  the  book  "  upside  down,"  he  will  see 
the  three  views  of  the  cube  in  the  relative  positions  they 
would  occupy  were  the  drawing  made  by  the  third-angle 
method.  It  will  be  seen  that  the  effect  is  merely  that  of 
rendering  the  plan  uppermost  on  the  drawing.  The  front 
elevation  then  is  seen  below  the  plan,  while  the  side  eleva- 
tion, as  in  first-angle  projection,  is  at  the  side  of  the  plan. 
In  the  case  of  a  simple  solid  like  the  cube,  the  use  of  differ- 
ent angles  of  projection  has  but  slight  effect,  but  when  a 
more  complicated  solid  is  represented  on  the  drawing,  it 
becomes  necessary  for  the  one  that  would  "  read  "  the  draw- 
ing to  know  which  angle  has  been  used  by  the  draftsman. 


58  PRACTICAL    PROJECTION  §  15 

Note  that  in  Fig.  23  the  secondary  projectors  are 
described  from  i^  as  a  center,  the  lines  of  the  cube  being 
used  as  base  Hues,  as  mentioned  in  Art.  39.  When  this 
short  method  is  adopted  in  the  case  of  secondary  project- 
ors, the  center  from  which  they  are  described  must  be 
located  at  that  intersection  of  the  primary  projectors  near- 
est the  two  views  between  which  secondary  projectors  are 
drawn.  A  further  illustration  of  this  will  be  given  in  con- 
nection with  a  later  drawing. 

66.  Iliciaen  Sui-faces :  How  Indicated. — When  the 
form  of  a  solid  is  such  that  in  any  view  a  smaller  surface  is 
hidden  by  a  larger,  the  smaller  surface  is  not  shown  in  that 
view,  although  frequently  its  outline  may  be  defined  by 
dotted  lines  on  the  drawing.  This  applies  also  to  projections 
in  which  two  or  more  solids  are  shown  in  positions  such  that 
some  of  their  surfaces  are  completely  or  partially  hidden  by 
other  surfaces  nearer  the  eye  of  the  observer.  Only  such 
surfaces  as  receive  the  lines  of  sight  directly  from  the  eye  of 
the  observer  are  shown  in  a  view  by  full  lines,  although,  as 
mentioned  above,  the  outline  of  such  other  surfaces  as  it 
may  be  desirable  to  show  in  a  drawing  may  be  indicated  by 
dotted  lines. 

67.  Facility  in  Reading  DraMangfS  Acquired  Only 
l>y  Practice. — The  reading  of  working  drawings  is,  there- 
fore, a  comparatively  easy  matter,  if  the  student  will  resolve 
each  portion  of  the  object  represented  into  its  respective 
surfaces  and  look  for  the  various  outlines  as  they  are  shown 
in  the  different  projections.  If  this  is  found  a  difficult  task, 
the  surfaces  may  be  further  resolved  into  lines  and  points, 
whose  respective  positions  may  then  be  located  in  each  view 
shown.  It  is  not  to  be  expected  that  the  position  of  every 
surface  in  a  complicated  drawing  will  be  seen  by  the  beginner 
at  a  single  glance — an  expert  seldom  acquires  such  profi- 
ciency— but  as  "practice  makes  perfect,"  the  student  may 
easily  accustom  himself,  by  careful  study  of  the  various 
positions    of    the    surfaces    composing    the    solids    that    are 


§  15  PRACTICAL    PROJECTION  59 

projected  in  the  following  problems,  to  the  more  or  less 
complicated  projections  found  in  the  various  mechanical 
and  architectural  journals,  in  shop  drawings,  or  in  such 
other  projection  drawings  as  are  within  his  reach. 

68.  The  Center  Ldne. — It  has  been  found  convenient, 
when  making  projections  of  objects,  to  make  use  of  a  line 
that  is  imagined  to  pass  through  the  central  portion  of  the 
solid  as  it  is  shown  in  any  plan  and  elevation.  Such  a  line 
is  called  a  cetiter  line,  and  in  many  projections  it  is  inked  in 
when  the  drawing  is  finished,  since  it  frequently  affords  a 
convenient  means  of  indicating  certain  positions  of  the  fig- 
ure, besides  assisting  in  the  location  of  the  several  surfaces 
of  the  solid  in  the  different  views.  This  line,  however,  is 
central  only  in  its  relation  to  the  object  of  which  the  draw- 
ing is  a  representation,  and  not  in  relation  to  the  planes  of 
projection.  This  may  be  better  understood  by  considering 
the  center  line  as  the  projection  of  an  imaginary  surface  (or 
plane)  that  passes  through  the  central  portion  of  the  figure. 
It  is  generally  represented  in  those  views  only  in  which  that 
imaginary  surface  can  be  shown  in  one  line,  or,  as  we  have 
said  before,  as  if  "on  edge."  Thus,  in  the  right  view  pro- 
jected in  Fig.  23,  the  lines  zu  x  a.vA  y  s  are  center  lines,  rep- 
resented on  the  drawing  by  the  broken-and-double-dotted 
lines  shown  in  Fig.  23.  The  practical  use  of  the  center  line 
will  be  illustrated  in  the  succeeding  problems  by  the  pro- 
jection of  solids  into  various  positions. 


PROBLEM    12 

(59.  To  dra>v  tli<?  projection  of  a  s'iven  solid,  sev- 
eral positions  beiiij^  indicated. 

The  solid  for  the  projections  of  this  problem  is  shown 
in  perspective  in  Fig.  24.  It  is,  as  the  figure  shows,a  pent- 
agonal prism;  that  is,  a  solid  whose  ends  are  pentagons  and 
parallel  to  each  other  and  whose  sides  are  parallelograms. 
The    dimensions   are    given    in    Fig.  24;    three    projections 


60 


PRACTICAL    PROJECTION 


§15 


are  -to  be  drawn,  each  showing  the  solid  in  a  different 
position;  each  projection  will  be  complete 
in  a  set  of  views  consisting  of  a  plan  and 
front  and  side  elevations.  Each  set  of  pro- 
jections is  to  occupy  one  of  the  remaining 
spaces  on  the  plate,  the  prism  being  shown 
in  the  positions  indicated  by  the  following 
cases : 

Case  I. — /;/  an  upright  position^  the  side 
nearest  the  observer  being  parallel  to  V  P. 


Fig.  24 


The  projections  showing  this  position  of 
the  prism  may  be  readily  drawn  by  the 
student  from  the  explanations  and  instructions  already 
given.  The  plan,  which  is  a  pentagon  with  f-inch  sides, 
should  be  drawn  first,  according  to  instructions  given 
in  Geometrieal  Drawing.  Draw  the  center  lines  as  in 
Fig.  23  and  ink  them  in,  completing  the  drawing  as  shown 
on  the  plate.  All  lines  that  represent  the  intersection 
of  surfaces — i.  e. ,  the  edges  of  the  prism — and  that  intercept 
the  lines  of  sight  directly  from  the  eye  of  the  observer  are 
to  be  represented  by  full  lines  on  the  drawing.  All  hidden 
edges  are  to  be  shown  by  dotted  lines. 

Case  II. — In  a  horizontal  position,  the  upper  side  being 
parallel  to  H  P,  and  the  ends  of  the  prism  parallel  to  y  P  in 
the  side  elevation. 

In  this  case,  which  differs  from  Case  I  only  in  the  posi- 
tion of  the  prism,  the  side  elevation  should  be  drawn  first. 
It  may  here  be  mentioned  that,  in  drawing  different  views 
of  objects  in  right  positions,  the  question  as  to  which  view 
is  drawn  first  is  merely  a  matter  of  convenience,  depending 
on  the  form  of  the  object  represented.  Thus,  in  Case  I 
the  plan  is  drawn  first ;  in  this  instance  it  is,  however,  more 
convenient  to  draw  the  side  elevation  first.  In  this  drawing 
it  is  only  in  the  plan  and  the  side  elevation  that  the  center  line 
can  be  shown,  since  an  edge  view  of  the  plane  represented 
would  not  be  given  in  the  front  elevation. 


§  15  PRACTICAL    PROJECTION  61 

Case  III. — ///  a  rightly  inclined  position,  the  angle  of 
inclination  of  the  center  line  {and  consequently  of  the  prisni) 
toW9  being  75°. 

Explanation. — The  sides  of  the  prism  are  in  the  same 
relative  position  to  V  P  in  the  front  elevation  as  in  Case  I. 
(See  the  plate.)  The  projection  of  solids  to  inclined  posi- 
tions is  accomplished  in  the  same  manner  as  the  projection 
of  surfaces  to  similar  positions  in  preceding  problems. 
Practical  use  may  be  made  of  the  center  line  in  this  drawing; 
this  line  may  be  shown  in  the  front  elevation,  but  cannot 
be  continued  to  the  plan  from  that  view,  since  it  is  clear 
that  its  true  position  may  not  be  shown  there  in  one  line. 

Construction. — First  draw  the  line  iu  x  in  that  part  of 
the  space  devoted  to  the  front  elevation  and  at  an  angle 
of  75°  to  the  base  line.  The  line  w,i'is  the  center  line  of  the 
front  elevation.  Next,  copy  the  front  elevation  of  Case  I  in 
the  same  relative  position  on  this  line,  thus  producing  the 
rightly  inclined  front  elevation.  Draw  primary  projectors 
vertically  downwards  from  all  points  of  this  front  elevation, 
and  intersect  them  with  horizontal  primary  projectors  drawn 
from  the  plan  of  Case  I,  in  the  same  manner  as  the  plan  of 
Problem  6  on  the  preceding  plate  was  produced.  Next, 
draw  the  outline  of  the  plan  by  connecting  the  intersections 
of  projectors  that  have  been  drawn  from  corresponding 
points  in  each  of  the  two  views.  Project  the  side  elevation 
by  means  of  secondary  projectors  described  from  the  inter- 
section of  the  upper  and  right-hand  primary  projectors,  that 
is,  at  O  on  the  plate  (see  Art.  65).  When  drawing  second- 
ary projectors  for  solids  whose  surfaces  do  not  extend  to 
the  outer  primary  projectors  in  the  adjacent  views,  note 
that  the  primary  projector  must  be  produced  as  at  Oy 
and  <9^. 

Case  IV. — ///  an  obliquely  inclined  position,  with  the  center 
line  at  an  angle  of  J^o°  to  M  P  and  15°  to  H  P,  the  upper  side 
being  in  such  a  position  that  a  full  viezv  ivill  be  shoivn  of  its 
upper  and  loiver  edges. 

M.  E.    V.—I6 


62 


PRACTICAL    PROJECTION 


§15 


Explanation. — As  in  Problem  10,  where  a  surface  was 
projected  to  an  obliquely  inclined  position,  so  in  this  prob- 
lem some  preliminary  work  must  be  done  on  another  piece 
of  paper.  These  preliminary  projections  are  shown  in 
Fig.  25,  of  which  (a)  is  the  right  plan  and  elevation  and  (d) 
is  the  rightly  inclined  drawing. 

Construction. — On  a  separate  piece  of  paper  construct 
the  projections  shown  at  (a),  Fig.  25;  copy  the  elevation 
produced  at  (a)  in  such  a  position  at  (d)  that  the  angle  of 
the  center  line  ic  x  is  15°  to  H  P,  as  required  by  the  condi- 
tions of  the  problem.  Next,  project  the  plan  in  {b)  from 
the  plan  of  {a),  in  connection  with  the  elevation  of  {b),  as 
indicated  by  the  primary  projectors  drawn  from  these  views. 
Redraw  the  plan  of  {b)  at  (f)  and  give  its  center  line  yz 

w 


Fig.  25 


the  required  angle  of  45°  to  V  P.  Next,  produce  the  eleva- 
tion in  {c)  by  drawing  primary  projectors  from  the  elevation 
in  {b)  and  intersecting  them  by  primary  projectors  drawn 
from  corresponding  points  in  the  plan  in  {c).  These  opera- 
tions are  precisely  similar  to  those  used  in  producing  the 
obliquely  inclined  views  of  the  surface  in  Problem  10;  and 
if  the  extra  piece  of  paper  is  laid  over  the  drawing  in  such  a 
manner  as  to  leave  the  lower  right-hand  space  exposed,  the 


/\  /     /     /  /T 


PRDJEC1 


PROBLEM  13. 
Ca5e/. 


-^-^ 


PROBLEM  14-. 
Case/. 


PROBLEM  13. 
CaseZ. 


PROBLEM  14. 
CaseS. 


JUNK  25J8S3 


Copyright,  1899,  by  THE  COI 
All  right! 


RV    ENGINEKR   COMIANY. 
ervpd. 


JOHAJ  3M/TH,  CLASS A/?^S2S. 


§  15  PRACTICAL    PROJECTION  63 

drawings  shown  in  Fig.  25  at  (c)  may  be  projected  directly 
to  their  proper  position,  thus  producing  the  plan  and  front 
elevation  required.  The  side  elevation  is  then  projected  by 
means  of  secondary  projectors  described  from  the  center  O, 
as  shown  on  the  plate.  It  is  thus  shown  that  oblique  views 
of  solids  are  projected  by  the  same  methods  as  those  used  in 
the  case  of  surfaces. 

Note. — The  appearance  of  this  plate  will  be  improved  if  the  base 
line  of  the  front  elevation  of  this  case  is  placed  |  inch  higher  than  in 
the  preceding  cases. 

TO.  Tlie  Axial  Iiine. — In  views  such  as  are  projected 
in  this  drawing — i.  e.,  obliquely  inclined  views — the  center 
line,  as  the  representation  of  the  central  plane  of  the  figure, 
can  be  shown  only  in  the  plan;  that  is,  in  the  position  of 
the  solid  shown  in  this  case.  The  position  of  the  solid  might 
be  such  that  the  center  line  would  be  shoAvn  in  one  of  the 
elevations,  or  possibly  not  in  any  right  view.  However,  it 
is  sometimes  represented  in  drawings  merely  as  a  central 
line,  and  not  as  the  representation  of  a  central  plane ;  it  may 
in  such  cases  be  projected  to  the  other  views  by  means  of 
points  located  at  convenient  distances  on  the  line.  It  is  then 
called  an  axial  line,  or  line  of  axis ^  and  represents  the  posi- 
tion of  the  axis  of  the  solid.  It  is  not  projected  in  the  cases 
of  the  preceding  problem,  since  the  figure  is  not  of  such  a 
form  as  to  demand  the  location  of  an  axial  line.  The  axial 
line  is  similar  in  its  use  to  the  center  line  and  is  represented 
on  drawings  by  the  same  kind  of  a  broken-and-dotted  line. 

In  order  to  avoid  confusion,  the  center  and  axial  lines  are 
usually  indicated  by  small  lettering  placed  conveniently 
near  one  end  of  the  lines ;  thus,  ' '  center  line"  or  ' '  axial  line. " 


DRAWI:NG    PT^ATE,  title  :     PROJECTIONS    III 

71.  Three  problems  are  to  be  drawn  on  this  plate,  each 
of  which  will  require  two  sets  of  projections.  Like  the  pro- 
jections of  Problem  12,  they  consist  of  a  plan  and  front  and 
side  elevations.  Each  set  of  projections  occupies  one  space 
on  the  drawing,  and  the  plate  is  divided  into  six  equal  spaces 


64 


PRACTICAL     PROTECTION 


§15 


by  a  single  horizontal  and  two  vertical  lines.  The  different 
cases  of  each  problem  are  drawn  in  the  same  vertical  divi- 
sion; thus,  Case  I  of  Problem  13  occupies  the  upper  left  hand 
space  and  Case  II  of  the  same  problem  is  directly  under  it. 
Cases  I  and  II  of  the  other  two  problems  on  the  plate  are  to 
be  placed  in  the  same  way  in  the  remaining  spaces. 


PROBLEM    13 

"t2.     To  dra^^  the  projections  of  a  cylindei*. 

The  method  of  projection  used  in  the  case  of  surfaces 
having  curved  outlines  has  been  shown  in  a  preceding 
problem.  Two  such  surfaces  are  presented  in  the  ends  of 
the  cylinder  shown  in  perspective  in  Fig.  26,  and  the  fig- 
ure also  gives  the  dimensions  of  the 
solid  as  it  is  to  be  drawn  on  the  plate. 
Since,  with  the  exception  of  those  at 
the  intersection  of  the  ends,  there  are 
no  edges  formed  by  the  curved  sides  of 
the  cylinder,  there  will  be  no  full  lines 
on  the  drawing  except  those  required 
to  show  the  ends  in  their  different 
positions  and  the  outline  of  the  sides. 
A  front  elevation  of  the  cylinder  in  the 
right  position  indicated  by  Fig.  26  is 
therefore  a  parallelogram,  the  length 
of  whose  horizontal  sides  is  equal  to  the  diameter  of  the 
circular  surfaces  at  the  ends  of  the  cylinder — the  vertical 
sides  equal  in  length  to  the  height  of  the  cylinder.  Its  entire 
dimensions  and  its  form  are  indicated  in  a  plan  and  front 
elevation,  and  there  is  no  need  of  making  any  further  draw- 
ings to  enable  the  mechanic  to  understand  the  shape  of  the 
solid  thus  represented. 

Case  I. —  When  the  cylinder  is  rightly  inclined. 

ExpLAXATiox. — In  this  drawing  the  cylinder  is  inclined  at 
an  angle  of  60°  to  H  P.  Rightly  inclined  views  of  solids  are 
often  drawn  by  a  method  somewhat  shorter  than  that  shown 


Fig.  36 


§  15  PRACTICAL    PROJECTION  65 

in  preceding  problems.  By  this  method,  temporary  views 
are  drawn  in  convenient  positions  on  the  paper  and  rightly 
inclined  views  are  projected,  as  shown  in  this  case.  Less  space 
is  reqtiired  for  the  drawing  and  a  saving  of  time  is  effected; 
the  principles  involved,  as  will  be  seen  from  the  following  con- 
struction, are  identical  with  those  of  the  preceding  problem. 

Construction. — First  draw  the  center  line  in  the  eleva- 
tion at  the  given  angle — as  A  />  in  the  drawing  on  the  plate. 
Describe  the  circle  shown  at  (///),  which  represents  a  full 
view  of  the  end  of  the  cylinder;  next,  draw  the  front  eleva- 
tion C  D  E  F  o\\  the  center  line  A  B  according  to  the  given 
dimensions.  Describe  a  circle  similar  to  (w)  at  {ji) ;  this  is  a 
temporary  view  of  the  end  of  the  cylinder  and  corresponds 
to  the  plan  of  the  prism  at  {ii).  Fig.  25.  Locate  a  conve- 
nient number  of  points  at  equal  distances  on  the  outline  of 
each  full  view  thus  drawn  at  (;//)  and  («).  Project  the  points 
of  (;//)  to  the  elevation  C  D  E  F,  and  thence  draw  primary 
projectors  vertically  downwards;  intersect  these  primary 
projectors  by  other  primary  projectors  drawn  horizontally 
from  similar  points  located  on  the  outline  of  the  full  view 
at  (;/).  Trace  the  outline  of  the  plan  thus  produced  through 
points  of  intersection  corresponding  to  those  on  the  full 
views.  The  temporary  full  views  {in)  and  (//)  may  then  be 
erased  from  the  plate.  Project  the  side  elevation  by  means 
of  secondary  projectors  described  by  the  arc  method,  thus 
completing  the  drawing. 

Case  II. —  Wlu'ii  tlic  cylinder  is  obliqncly  inclined. 

Explanation. — The  method  of  ]:)r()jecting  the  drawings 
required  for  this  case  is  similar  to  that  already  given  for 
oblique  views  of  surfaces  and  solids,  and  has  been  fully 
explained  in  Art.  60,  and  also  in  connection  with  Case  IV  of 
Problem  12.  A  right  view  is  first  drawn  [as  the  elevation 
and  full  view  {in)  of  the  preceding  case] ;  next,  a  rightly 
inclined  view.  The  rightly  inclined  plan  thus  drawn  is 
then  recopied  at  the  given  angle,  thus  producing  the  plan  of 
the  oblique  view;  from  this  plan,  in  connection  with  the 
rightly  inclined    elevation,  the  obliquely  inclined  elevation 


66 


PRACTICAL    PROTECTION 


§15 


is  projected.  The  rightly  inclined  plan  and  elevation 
having  been  drawn  in  Case  I  of  this  problem,  the  plan  there 
shown  may  be  redrawn  for  the  plan  of  this  case. 

CoxsTRUCTiox. — On  a  separate  piece  of  paper  reproduce 
the  plan  and  front  elevation  of  Case  I  and  fasten  this  paper 
b)'  thumbtacks  to  the  drawing  board,  towards  the  left  of  the 
space  used  for  this  case.  Next,  redraw  the  plan  of  Case  I 
in  its  proper  place  on  the  plate  for  this  case,  and  in  such  a 
position  that  the  line  G  H  oi  Case  I  forms  an  angle  of  60° 
with  the  base  line  of  the  front  elevation,  as  shown  at  G'  H' 
on  the  plate.  Then  produce  the  front  elevation  by  drawing 
primary  projectors  upwards  from  the  plan  and  intersecting 
them  by  similar  projectors  drawn  horizontally  from  the 
rightly  inclined  elevation  on  the  attached  sheet,  which  may 
then  be  removed.  Trace  curves  through  the  points  thus 
projected  and  draw  the  tangential  lines,  as  previously 
described  and  as  shown  at  C  D'  P  E'  on  the  plate.  Project 
the  side  elevation  as  in  preceding  problems,  taking  special 
care  to  project  fro'm  similar  points  in  each  view. 


3. 


PROBLEM    14 

To  dra%v  tlie  projections  of  a  hexagonal  pyi*amid. 

A  pyramid  is  a  solid  whose  base  is  a  polygon  and  whose 
sides  are  triangles  uniting  at  a  common  point  called  the 
vertex.  The  pyramid  for  the  projec- 
tions of  this  problem  is  shown  in 
Fig.  27,  where  its  dimensions  are  clearh* 
indicated.  Since  these  drawings  are 
very  easy  and  are  constructed  in  a  man- 
ner similar  to  those  of  preceding  prob- 
lems, definite  instructions  are  omitted, 
and  the  student  is  expected  to  be  able 
to  complete  the  drawings  by  the  aid  of 
the  brief  explanations  that  follow. 

Case  I. —  When  the  pyramid  is  in  a 
Fig.  27  right  position. 


§15 


PRACTICAL    PROJECTION 


67 


Explanation. — The  plan  of  this  projection  is  most  con- 
veniently drawn  tirst,  a  circle  If  inches  in  diameter  being 
described  from  (9  as  a  center,  as  shown  on  the  plate.  The 
edges  of  the  pyramid  are  then  drawn :  a  horizontal  diameter 
and  two  diameters  at  angles  of  60°  with  the  first  repre- 
sent the  upright  edges;  chords  of  the  arcs  thus  designated 
are  then  drawn,  and  the  plan  of  the  pyramid  is  complete. 
Next  draw  the  center  lines  A  B  and  C  D\  set  off  the  height 
of  the  pyramid  on  the  line  A  B  and  complete  the  front  ele- 
vation by  the  aid  of  primary  projectors,  as  shown  on  the 
plate,  the  side  elevation  being  projected  as  in  former 
problems. 

Case  II. —  When  the  pyramid  is  in   an  obliquely  inclined 
position. 

Explanation. — The  angles  of  inclination  in  the  projec- 
tions of  this  case  are  60°  to  H  P  and  45°  to  V  P.     Preliminary 


Fio.  28 


drawings  are  required  on  separate  paper,  as  shown  in  Fig.  28, 
first,  as  at  {a),    showing  a  right  view  of  the  pyramid;  and 


68  PRACTICAL    PROJECTION  §  15 

second,  as  at  {b),  showing  a  rightly  inclined  view,  the  angle 
of  inclination  (of  the  center  line)  being  60°  to  H  P.  The 
plan  produced  at  {b)  is  then  copied  on  the  plate  in  such  a 
position  that  its  axial  line  will  make  the  required  angle, 
viz.,  45°  to  V  P,  as  shown  by  the  line  zc  ,r  at  (c),  Fig.  28. 
The  front  elevation  is  then  projected  as  in  Case  II  of  the 
preceding  problem,  that  is,  by  vertical  primary  projectors 
drawn  from  the  plan  in  (r),  intersected  by  horizontal  pri- 
mary projectors  drawn  from  the  elevation  in  (/?).  The  pro- 
jection of  the  side  elevation  by  the  arc  method  of  secondary 
projectors  is  also  similar  to  the  preceding  projections,  as 
will  be  seen  from  an  inspection  of  the  plate. 


PROBLEM    15 

74.     To  draw  the  projections  of  a  cone. 

Thercwr  is  a  solid  that  may  be  produced  by  the  revolution 
of  a  right-angled  triangle  around  one  of  its  sides  as  an  axis. 
Its  base,  therefore,  is  a  circle,  and  its  curved  surface  tapers 
uniformly  towards  a  point  at  the  top  called  the  vertex,  or 
apex.  Like  the  cylinder,  its  entire  form  and  dimensions 
are  presented  in  a  plan  and  a  single  elevation  showing  a  right 
view  of  the  cone.  The  cone  for  the  projections  of  this  prob- 
lem is  shown  in  perspective  in  Fig.  29,  which  gives  the 
dimensions  that  the  cone  is  to  present  on  the  plate.  The 
methods  used  are  precisely  similar  to  those  used  in  the  case 
of  the  hexagonal  pyramid  in  the  preceding  problem. 

Case  I. — /;/  a  rightly  inclined  position. 

Explanation. — In  order  to  produce  the  rightly  inclined 
front  elevation,  a  construction  similar  to  that  used  in  Case  I 
of  Problem  13  is  here  used.  The  drawing  differs  from  that 
projection  only  in  the  form  of  the  solid.  The  angle  of 
inclination  in  this  case  is  50°  to  H  P. 

Construction. — Draw  the  center  line  A  B  (see  the  plate) 
at    the  given  angle,  that    is,  50°  to  the  horizontal.      Next, 


§15 


PRACTICAL    PROJECTION 


69 


pro- 


construct  the  triangle  representing 
the  elevation  of  the  cone  and  describe 
the  circle  at  (;//) — a  temporary  full 
view  of  the  base.  Describe  a  similar 
circle  at  («),  also  a  full  view  of  the 
base,  and  locate  a  convenient  number 
of  points  on  the  outline  of  each  full 
view — in  this  case  eight — as  shown  at 
rt,  b,  c,  etc.,  on  the  plate.  Project  the 
rightly  inclined  plan  in  a  manner  pre- 
cisely similar  to  that  used  in  the  view 
of  the  cylinder  in  Case  I  of  Prob- 
lem 13.  Erase  the  temporary  views  (/;/)  and  (;/), 
ject  the  side  elevation  as  in  preceding  projections. 

Case  II. — ///  an  obliquely  inclined  position. 

ExPLAXATiox. — The  angle  of  inclination  to  H  P  is  the  same 
as  in  the  drawing  last  made,  and  the  plan  of  that  projection 
may  be  recopied  for  the  plan  of  this  case,  but  it  is  to  be 
drawn  on  the  plate  in  such  a  position  that  the  center  line  it'  x 
will  make  an  angle  of  45°  to  V  P.  The  plan  and  front  eleva- 
tion of  the  preceding  case  must  be  redrawn  on  separate 
paper  and  temporarily  fastened  over  the  drawing  towards 
the  left  of  the  space  required  for  this  case,  in  order  that  the 
projection  of  the  front  elevation  may  be  drawn.  As  this 
process' is  similar  to  that  used  in  preceding  constructions,  no 
further  explanation  will  be  given.  Complete  the  projections 
in  the  plan,  front,  and  side  elevations  as  shown  on  the  plate. 

Note. — When  inclined  views  are  drawn  of  solids  having  curved 
surfaces  (as  the  cylinder  and  the  cone),  the  circular  ends  should  first 
be  projected.  The  outline  of  the  curved  surface  is  then  represented 
as  tangent  to  the  base,  or  bases,  of  the  solid,  and  without  regard  to  the 
intersection  of  such  outline  with  any  given  point  on  the  base  outline. 

75.  Self-Heliance. — ^The  student  that  has  intelligently 
completed  the  projections  of  the  foregoing  problems  and 
has  made  frequent  use  of  the  imaginative  feature  of  this 
subject,  as  previously  explained  and  directed,  should  now 
possess  a  very  complete  knowledge  of  the  methods  of  pro- 
jection used  in  representing  plain  solids  in  various  positions. 


70  PRACTICAL    PROJECTION  §  15 

The  projection  of  irregularly  outlined  figures  has  not  been 
presented,  since  the  methods  are  identical  with  those 
already  shown.  The  student  should  acquire  a  degree  of 
self-reliance  in  this  work:  for  if  he  is  to  depend  on  having 
the  projection  of  every  conceivable  form  described  for  him, 
the  principles  governing  those  projections  will  become  a 
secondary  matter,  whereas  the  practical  draftsman  requires, 
above  all  else,  the  faculty  of  recognizing  the  principles  by 
which  to  define  and  project  the  various  forms  occurring  in 
the  course  of  his  work. 

Note. — The  student  should  understand  that  the  percentage  of 
marking  adopted  for  these  plates  is  based  on  the  degree  of  accu- 
racy in  which  the  projections  are  drawn  to  the  angles  of  inclina- 
tion, as  well  as  on  the  quality  of  neatness  attained  in  the  finish  of  the 
drawings. 

DRATTIXG    PI^\TE,   TITLE:    SECTIONS    I 

76.  I'se  of  Section  Di-a-svings. — A  section  drawing, 
as  previously  explained,  is  a  projection  of  a  portion  of  a 
solid,  in  a  view  where  the  solid  is  intersected  by  a  plane. 
This  plane — sometimes  called  a  cutting  plane — may  pass 
through  the  solid  in  any  direction;  that  portion  of  the  solid 
between  the  plane  and  the  observer  is  assumed  to  have  been 
removed.  Section  drawings  are  useful  in  many  ways,  for  by 
such  means  the  construction  of  interior  parts  of  objects 
may  be  shown.  It  is  desirable  in  many  cases  to  show 
some  particular  form  that  a  solid  of  peculiar  shape  pos- 
sesses at  a  place  where  it  cannot  be  presented  in  an 
exterior  view,  and  in  such  cases  sections  are  projected. 
The  section  is  considered  simply  as  a  surface  that  would 
appear  on  the  cutting  plane  in  any  view  of  the  object  pro- 
jected. The  outline  of  this  surface,  therefore,  will  depend  on 
the  form  of  the  object,  the  number  of  surfaces  intersected  by 
the  plane,  and  the  angle  of  inclination  of  the  cutting  plane. 

To  project  the  views  of  a  solid  in  which  a  new  surface  is 
thus  presented,  it  is  necessary  to  consider  the  various  sur- 
faces that  originally  composed  the  solid.  The  first  view 
drawn  is  always  that  in  which  the  cutting  plane  is  repre- 
sented as  oil  edge.     The  intersections  of  the  original  surfaces 


9  D 

f^J  (b) 

PROBLEM  /6. 


JUNE  25,  /393. 


Copyright,  1899,  by  The  Coli 


All  rights  ■ 


)N5- 


:/  /  M       J} 


PROBLEM   /^ 


■Y  Engineer  Company. 

rved. 


JOfYN  SMFTH,     CLAS3  NS  4529. 


§15 


PRACTICAL    PROJECTION 


71 


of  the  solid  with  the  cutting  plane  are  thus  shown;  and 
by  using  methods  of  projection  already  presented,  any  view 
of  the  section  may  then  be  drawn.  Those  portions  of 
the  solid  assumed  to  have  been  removed  are  not  shown  in  a 
section  drawing,  although  their  position  is  sometimes  indi- 
cated by  dotted  lines. 

77.  Sections  of  the  Sphere. — The  solid  that  presents 
the  most  simple  illustrations  of  sections  is  the  sphere,  or  globe. 
This  solid,  also  called  a  ball,  is  such  as  would  be  generated  by 
the  revolution  of  a  circle  around  its  diameter  as  an  axis. 

The  cylinder,  cone,  and  sphere  are  sometimes  called  the 
"three  round  bodies,"  or  the  solids  of  revolution.  It  will  be 
shown  later  that  their  imaginary  formation  by  such  revolution 
may  be  taken  advantage  of  by  practical  short  methods  of  pro- 
jection, the  principles  of  which  are  based  on  this  knowledge. 

A  full  view  of  any  section  of  a  sphere  is  a  circle.  If  the 
cutting  plane  passes  through  the  center  of  the  sphere,  the 
full  view  of  the  section  is  a  circle  whose  diameter  is  the  same 
as  the  diameter  of  the  sphere — or  the  great  circle  of  the 
sphere,  as  it  is  called.  If  the  cutting  plane  intersects  the 
sphere  in  any  other  way,  the  section  is  still  a  circle,  but  of 
smaller  diameter,  and  is  measured  from  a  view  in  which 
the  cutting  plane  is  shown  on  edge. 


78.     Sections    of 

drawing  of  the  cube 
referred  to  in  for- 
mer illustrations 
and  shows  a  vertical 
section.  It  will  be 
noticed  that  the 
view  of  the  section 
in  the  elevation  is 
the  same  as  the 
view  in  the  front 
elevation  in  Fig.  23.  ' 
This  is  always  the 
case      in      sections 


the    Cube. — Fig.   30  is  a  projection 


Fig.  30 


73 


PRACTICAL    PROJECTION 


15 


of  regular  prisms  where  the  cutting  plane  is  parallel  to 
the  ends  of  the  prism.  Fig.  31  represents  a  diagonal 
section  of  the  cube,  the  measurements  of  which  will  be 
apparent  to  the  student  from  an  inspection  of  the  drawing. 
Fig.  32  is  an  oblique  section,  in  which  but  three  sides  of  the 
cube  are  intersected  by  the  cutting  plane ;  in  this  figure, 
the  full  view  of  the  sectional  surface  is  projected.  Fig.  33 
represents  a  section  taken  at  a  still  different  angle  and  posi- 
tion of  the  cube.  ]]lien  a  cutting  plane  passes  through  a  solid 
having  parallel  sides,  in  anj'  direction  that  causes  it  to  inter- 
sect both  of  those  parallel  sides,  those  sides  are  shown  in  any 


Fig.  31 


viezv  by  parallel  lines.  Note  that  in  the  sectional  views  of 
Fig.  33  the  opposite  edges  of  the  surfaces  are  defined  by 
parallel  lines;  thus,  since  AB  and  CD  are  parallel  to  each 
other  in  the  plan  of  the  cube  in  Fig.  33,  so  A'  B'  and  CD' 
are  in  the  same  relation  to  each  other  in  the  side  elevation ; 
also,  A"  B"  and  C"  D"  in  the  full  view  of  the  section.  The 
same  is  also  true  oi  A  C  and  B  D,  as  may  be  seen  from  a 
comparison  of  the  views. 

79.     Ho^v  the  Cutting  Plane  Is  Represented. — When 
the    cutting    plane    is    shown    on    edge    in   a    view,    it    is 


15 


PRACTICAL    PROJECTION 


73 


usually  indicated  by  the  same  kind  of  a  broken-and-double- 
dotted  line  used  for  the  center  line  and  the  axial  line.  The 
use  of  this  line  for  these  three  purposes  is  somewhat  puzzling 
to  the  beginner;  but  as  the  student  is  now  able  to  read  pro- 
jection drawings,  he  can  readily  determine  which  purpose 
the  line  is  intended  to  serve.  It  is  customary,  however,  as 
already  mentioned,  when  center  and  axial  lines  are  used,  to 
mark  them  as  such  by  neat  lettering.  A  section  line  in  a 
complicated  drawing  is  usually  designated  by  a  letter  placed 


Fig.  32 

at  each  end  of  the  line,  to  which  reference  is  made  in  the 
following  manner:  If,  in  the  view  where  the  cutting  plane 
appears  as  a  line,  it  is  lettered  A-B,  the  full  view  of  the 
section  is  designated  as  a  "section  on  the  line  A-B." 

80.  The  problems  for  this  plate,  of  which  there  are 
four,  consist  of  the  projection  of  the  section  drawings 
indicated  in  tlie  accompanying  illustrations.  They  are  to  be 
reproduced  on  the  plate  by  the  student  to  the  dimensions 


T4 


PRACTICAL    PROJECTION 


§15 


given  on  each  figure.  The  cutting  plane  is  indicated  by  the 
line  A-B.  and  the  views  shown  may  be  understood  by  a  care- 
ful study  of  the  figures  on  the  plate  illustrating  each  problem. 
The  direction  of  a  certain  line  in  each  plan  is  changed  in  these 
views,  and  the  cross-section  is  accordingly  represented  as 
foreshortened  in  the  front  and  side  elevations.  A  view  is 
also  to  be  projected  in  which  a  full  view  of  the  sectional 


Fig.  33 


surface  will  be  seen.  The  plate  is  to  be  divided  into  four 
equal  spaces  b)'  horizontal  and  vertical  lines.  Problem  16 
is  to  occupy  the  upper  left-hand  space. 

The  student  is  recommended  not  to  refer  to  thfe  reduced 
copy  of  the  plate  more  frequently  than  is  necessary  to  enable 
him  to  fix  the  location  of  the  views  on  the  drawing;  he  should 
learn  to  depend  on  his  own  knowledge  of  projection. 


15 


PRACTICAL    PROJECTION 


75 


PROBLEM    16 

81.  To  project  sectional  views  of  an  octagonal 
prism,  the  cutting  plane  crossing  the  solid  at  an 
oblique  angle  and  leaving  a  portion  of  the  upper 
surface   intact. 

Explanation. — The  position  of  the  prism  and  the  angle  of 
the  cutting  plane  is  shown  in  Fig.  34.  THis  figure  is  to  be 
drawn,  as  at  (a)  in  the  left-hand  portion  of  the  space,  to  the 
size  required  by  the  dimension  figures.  The  plan  is  then 
copied  to  the  right,  as  at  (d),  but  in  a  relatively  different 
position,  as  will  be  seen  by  the  arrangement  of  the  letters 
and  the  direction  of  the  line  CD.  Thus,  the  edge  (7,  which 
in  the  plan  of  (a)  is  on  the  extreme  left  of  the  figure,  occu- 
pies a  position  nearer  the  lower  part  of  the  drawing  in  the 
plan  at  (d).  This  shifting  of  position  of  the  plan  may  be 
effected  by  describing  circles  circumscribing  the  octagons 
and  drawing  the  diameter  C D  at  an  angle  of  30°  to  V  P,  as 
shown  in  the  plan  at  (a).  This  diameter  is  then  drawn  in 
a  vertical  position  in  the  second  plan  (d),  after  which  the 
arc  D  a,  as  measured  on  the  first, 
may  be  set  off  with  the  dividers  on 
the  second,  plan.  The  projections 
of  the  front  and  side  elevations  are 
then  made  in  the  regular  way.  The 
projection  of  the  full  view  is  accom- 
plished by  drawing  projectors  at  right 
angles  to  the  cutting  plane  A  B.  At 
a  convenient  distance,  as  shown 
at  (<•),  draw  the  center  line  a"  j'\  and 
from  this  line  set  off  with  the  dividers 
the  distances  from  the  line  a  c  as 
found  in  the  first  plan.  As  similar 
positions  have  corresponding  letters 
in  the  different  views  on  the  plate, 
the  student  will  have  no  difficulty  in  D' 
recognizing  the  method  of  transfer, 
it  being  the  same  as  that   mentioned  ^"''  ** 

in  Art.  53,     Observe  that  the  sides  of  the  section  b'  c'  and 


76 


PRACTICAL    PROJECTION 


§15 


g'  o\  and  ^"  r"  and  g"  o'\  are  parallel  in  every  view  shown, 
since  the  sides  b  c  and  go{f)  are  parallel,  as  shown  in  the 
plan  of  the  solid.  The  same  is  true  of  h g  and  r  t?  (</),  as  seen 
at  h" g"  and  c"  o"  in  {c)  (see  Art.  78). 

The  student  will  finish  the  drawing  on  the  plate  in  as 
complete  a  manner  as  in  the  drawing  of  the  cube  in  Fig.  30, 
but  omitting  all  reference  letters.  The  surface  of  all  sec- 
tions in  each  view  is  to  be  cross-hatched,  as   in  Figs.  30-33. 


82.     To 
pjTaiiiid. 


PROBLEM    17 

project    sectional    vie"\vs 


of    a    liexagronal 


The  dimensions  and  position  of  the 
pyramid  and  the  angle  of  the  cutting 
plane  are  indicated  in  Fig.  35.  Any  sec- 
tion of  a  pyramid  taken  at  right  angles 
to  the  axis  of  the  solid  is  a  polygon 
having  an  outline  similar  to  the  base 
of  the  pyramid.  The  polygon  repre- 
senting the  section  of  any  pyramid 
thus  intersected  varies  in  size  as  the 
cutting  plane  passes  through  the 
pyramid  at  a  point  on  the  axis  nearer 
the  base  or  the  vertex  of  the  pyra- 
mid. If  the  cutting  plane  passes 
through  the  vertex  and  the  base, 
or  through  two  sides  and  the  base  of 
the  pyramid,  the  section  is  a  triangle. 
Fig.  35  Any  other  section  is  a  polygon  having 

sides  unequal  in  length,  but  equal  in  number  to  the  sides  of 
the  polygon  forming  the  base  of  the  pyramid. 

Explanation. — The  projections  of  this  solid  and  its  sec- 
tions do  not  differ  materially  from  those  of  the  preceding 
problem.  The  change  of  position  in  the  two  plans  is  accom- 
plished by  means  of  the  circumscribing  circle,  the  diameter 
in  the  first  plan  being  drawn  at  an  angle  of  45°  and  the 


§15 


PRACTICAL    PROJECTION 


77 


arc  C  a  being  measured  in  a  similar  manner.  This  drawing 
may  be  more  accurately  made  if  the  edges  of  the  i)yramid 
are  continued  by  dotted  lines  to  the  vertex  O  in  the  fnmt 
elevations,  as  shown  on  the  plate.  A  comparison  of  the 
•reference  letters  in  the  different  views  on  the  plate  will  assist 
the  student  in  the  work  of  drawing  these  projections,  as  each 
line  shown  is  similarly  lettered  in  all  views. 


PROBLEM    18 

83.     To  project  sectional  vIcavs  of  a  cylinder. 

The  cylinder  possesses  some  points  of  similarity  to  the 
regular  prism  with  respect  to  its  section,  viz.,  a  section  par- 
allel to  its  ends  has  the  same  outline 
as  the  ends,  while  a  section  parallel 
to  its  axis  is  a  parallelogram.  An 
oblique  section  of  the  cylinder  is,  how- 
ever, as  will  be  observed  during  the 
projection  of  this  problem,  an  ellipse ; 
and  it  will  further  be  noted  that  a 
certain  view  of  an  ellipse  is  a  circle.         . 

ExPL.\x.A.Tiox. — The  position  of 
the  cylinder  and  the  angle  of  the 
cutting  plane  is  shown  in  Fig.  3G. 
Make  the  edge  a-a'  of  the  elevation 
at  {a)  yV  i'lch  long.  The  plan  of  the 
cylinder  is  first  divided  into  an  equal 
number  of  spaces  (in  this  case  13),  ^ 
the  points  being  lettered  a,  /;,  r,  etc., 
as  shown  on  the  plate.  These  points 
are  then  assumed  to  represent  edges,  as  in  the  case  of 
the  octagonal  prism,  and  are  projected  to  the  elevation, 
their  respective  positions  there  being  indicated  by  similar 
letters;  thus,  the  edge  a-a'  in  the  elevation  represents  the 
edge  a  as  shown  on  the  plan.  The  change  of  position  in 
the  second  plan  is  effected  by  drawing  the  vertical  diam- 
eter /  /",  which  is  shown  as  CD  in  the  first  i)lan  at  an  angle 


Fig.  .30 


M.  E.     l'.-r7 


78 


PRACTICAL    PROJECTION 


15 


of  SC  to  V  P.  The  position  of  all  points  is  then  noted  in  a 
manner  similar  to  the  two  preceding  problems  and  the  pro- 
jections are  completed  as  heretofore.  After  tracing  the 
curve  of  the  ellipses  through  the  points  located  in  the  differ- 
ent views,  the  sections  are  indicated  by  cross-section  lines, 
as  before  directed. 


84. 


PROBLEM    19 

To    project    views    of    an    irregularly    formed 


solid ;    also,  to  project  a  sectional  vie^v  from  a  given 
cutting  plane. 

Explanation. — This  solid,  shown  in  perspective  in  Fig.  37, 
may  be  described  as  a  transition  piece,  that  is,  a  form  used 
to  connect  openings  or  outlines  unlike  in  shape.  Such 
solids  are  frequently  used  in  the  sheet-metal  trades, 
particularly  in  boiler  and  pipework.  The  dimensions  of 
this  solid  are  better  shown  in  the  projection  at  Fig.  38. 
Its  upper  base  is  a  circle  1^  inches  in  diameter,  while  the 
lower  base  is  an  oval,  which 
may  be  drawn  by  the  method 
shown  in  Geometrical  Dratv- 
ing.  The  end  circle  of  the 
lower  base  is  described  with 
a  radius  of  1  inch  from  a  cen- 
ter located  at  B^  Fig.  38,  the    j 


Fig.  37 


distance  between  the  centers  of  the  upper  and  lower  bases, 
as  measured  on  the  plan  between  A  and  i/.  Fig.  38,  being 
f  inch.     The  vertical  height  of  the  solid  is  1^  inches. 


§  15  PRACTICAL    PROJECTION  79 

The  plan  and  front  and  side  elevations  are  drawn  as 
shown  on  the  plate,  and  since  the  methods  of  projection 
are  no  different  from  preceding  problems  representing 
regular  solids,  definite  instructions  for  this  work  are 
omitted.  Particular  attention  is  directed  to  the  method 
of  finding  the  outline  of  the  section,  as  its  application  to 
various  processes  of  patterncutting  is  of  great  importance. 
The  direction  of  the  cutting  plane  is  shown  in  the  side 
elevation  at  (a)  by  the  line  A  B  on  the  plate.  The  plan 
and  elevation  is  to  be  redrawn  on  the  plate  in  the  position 
shown  thereon  at  {b),  and  the  full  view  of  the  section  is 
projected  towards  the  upper  part  of  the  drawing.  As  it  is 
desired  to  show  a  section  through  a  certain  portion  of  this 
solid,  it  is  first  necessary  to  locate  a  number  of  points  on 
the  line  that  represents  the  cutting  plane.  Since  the  solid 
presents  no  edges  that  intersect  this  plane,  a  number  of  lines 
are  to  be  assumed  as  drawn  on  the  curved  surface  of  the  solid. 

Construction. — First  draw  a  horizontal  line  through  the 
central  portion  of  the  plan,  as  in  n  in  the  drawing  on  the 
plate;  this  divides  the  figure  into  symmetrical  halves. 
Next,  by  spacing,  divide  the  outline  of  each  base  into  the 
same  number  of  equal  parts,  as  at  a,  b,  c,  etc.  on  the  upper 
base  and/,  q,  r,  etc.  on  the  lower  base.  Project  the  points 
thus  located  to  the  elevation  and  draw  connecting  lines  in 
both  views,  as  shown.  Across  the  elevation  draw  the  line  CD, 
representing  the  cutting  plane,  and  project  the  intersections 
of  this  line  at  1,  3,  3,  etc.  to  the  plan,  producing  each  line 
thus  drawn  until  it  meets  the  horizontal  line  ///  //.  The  full 
view  of  the  sectional  surface  may  now  be  drawn  as  follows: 
Draw  projectors  vertically  upwards  from  the  points  1,  '3,  3,' 
etc.  in  the  elevation,  and  at  a  convenient  height  from  that 
view  draw  the  horizontal  line  ;//'  n'.  The  width  of  the  sur- 
face, as  measured  on  each  line,  may  now  be  set  off  with  the 
dividers,  as  directed  in  Art.  53;  thus,  take  the  space  ,v  j' 
from  the  plan  and  transfc-r  it  to  corresponding  positions 
at  x'  y  x'  on  the  same  line  in  the  full  view.  Transfer,  in 
like  manner,  the  length  of  each  vertical  dotted  line  shown 


80 


PRACTICAL    PROJECTION 


i5  15 


in  the  plan  to  the  full  view  and  trace  an  irregular  curve 
through  the  points  thus  located,  completing  the  projection. 
This  curve  may  also  be  traced  through  the  plan  and  there 
indicated  by  short  cross-hatching,  while  the  full  view  of  the 
sectional  surface  is  designated,  as  heretofore,  by  cross-sec- 
tion lines  in  the  manner  shown  on  the  plate. 

So.     Pi*actieal  Metliod  of  Representing  Certain  Sec- 
tions.— In   working  drawings,  particularly  those    made   for 


Fig.  .39 

the  execution  of  sheet-metal  work,  a  section  drawing  is 
often  made  directly  over  another  view,  and  the  lines  indi- 
cating the  section  are  distinguished  by  short  cross-hatching, 
as  shown  on  the  plate  in  the  plan  of  the  preceding  problem. 
Short  portions  of  sectional  views  are  also  frequently  shown 
in  this  way  on  working  drawings,  in   order  to  indicate  the 


form,  or  profile,  of  moldings,  the  different  planes  of  adjacent 
surfaces,  or  the  different  levels  at  which  certain   notations 


SECTIC 


PROBLEM  ZO. 

Case  /. 
A 


PROBLE/-i  ao.  Case  3. 


JUNE  2S.J&93. 


Copyright,  1S99,  by  THE  Coi 

All  righti 


Case  a 


3 
PROBLEM  SI. 


:rv  Kngineer  Companv, 
served. 


JOHN  5/^ITH,  CL  A55  N°  4S29. 


§15 


PRACTICAL    PROJECTION 


81 


are  made.  Thus,  in  Fig.  39,  which  shows  a  gable  finish, 
the  section  at  A  is  the  profile,  or  "stay,"  of  the  molding, 
while  the  section  at  B  indicates  that  the  panel  shown  is  a 
s?//ik  panel. 

In  detail  drawings  (mere  projections  drawn  full  size)  for 
decorative  sheet-metal  work,  sections  are  frequently  shown 
at  different  portions  of  the  drawing,  as  in  Fig.  40,  an 
inspection  of  which  will  enable  the  student  fully  to  compre- 
hend the  character  of  the  various  parts  without  the  aid  of 
another  view.  Sections,  therefore,  properly  understood  and 
used,  may  be  employed  by  the  draftsman  in  many  ways, 
and  are  frequently  a  means  of  shortening  the 
labor  of  drawing  intricate  projections. 

The  lines  representing  the  sides  of  the 
octagonal  shaft  in  Fig.  41  are  broken  in  the 
lower  part  of  the  figure,  this  being  a  means 
of  indicating  that  the  full  length  is  not  shown 
in  the  drawing.  Reference  to  the  dimension 
figures  gives  the  reason,  it  being  obviously 
unnecessary  to  make  a  drawing  extending 
the  full  length  of  a  simple  form. 


1 

K 

Ai^^^ii 

Fig.   11 


Besides   their 


DRAWIT^G    PIRATE,    TITLE: 
SECTIONS    II 

86.  Conic  Sections. — The  divisions  on 
this  plate  are  the  same  as  on  the  preceding 
plate.  The  two  problems  to  be  constructed 
are  those  relating  to  sections  of  the  cone, 
use  in  many  calculations  of  the  arts  and  higher  sciences, 
conic  sections  are  of  great  value  to  the  architectural  and 
mechanical  draftsman,  for  the  ctirves  thus  developed  pos- 
sess great  beauty  and  symmetry,  and  when  used  in 
moldings  present  pleasing  architectural  effects  of  light  and 
shade. 

The  full  view  of  the  section  of  a  regular  cone  made  by  a 
cutting  plane  parallel  to  its  base  is  z.  circle ;  if  the  cutting 
plane  passes  through  the  vertex  and  the  base  of  the  cone, 
the    section    is   a   triangle ;    if  the    cutting    plane  is  at   an 


82 


PRACTICAL    PROJECTION 


15 


oblique  angle  to  the  base,  which  angle  is  less  than  the  angle 
made  by  the  elements  of  the  cone  with  the  base,  the  section 
is  an  ellipse ;  if  the  cutting  plane  is  parallel  to  any  line 
drawn  on  the  convex  surface  of  the  cone  from  the  base  to 
the  vertex — that  is,  parallel  to  any  element  of  the  cone — the 
section  is  a  parabola ;  if  the  cutting  plane  is  at  any  angle 
(but  not  passing  through  the  vertex  of  the  cone)  greater 
than  the  angle  which  the  elements  of  the  cone  make  with 
the  base,  the  section  is  an  hyperbola. 


87.  Elements  of  tlie  Cone. — In  the  construction  of 
Problem  19,  where  it  was  desired  to  project  a  section  of  a 
solid  whose  curved  surface  presented  no  edges,  it  was  found 
necessary  to  assume  lines  on  that  surface  in  order  to 
establish    points    of    intersection    Avith    the    cutting    plane. 

This  process  must  be  followed 
with  all  solids  whose  surfaces  are 
curved,  and  in  the  case  of  the 
cone  the  lines  thus  assumed  have 
a  further  use,  as  will  appear  later. 
The  manner  in  which  these  lines 
are  located  on  the  surface  of  the 
regular  cone  is  of  special  impor- 
tance and  is  as  follows:  The  out- 
\y<C  line  of  the  base  in  the  plan  is  first 
divided  by  spacing  it  into  a  number 
of  equal  parts  (16  in  the  plan  of 
Fig.  42).  The  points  thus  located 
are  then  projected  to  a  view  in 
which  the  cone  is  represented  as  in 
the  elevation  of  Fig.  42 — that  is,  a 
right  view — and  lines  are  then 
drawn  from  these  points  to  the 
vertex.  They  are  also  shown  on 
the  plan  in  the  same  relation, 
^'*^-  ^  each    line  being    represented    as    a 

radius  of  the  circle  at  the  base  of  the  cone.     These  lines  are 
called  the  elements  of  the  cone. 


§  15  PRACTICAL    PROJECTION  83 

In  such  a  view  of  a  cone  as  presented  in  the  elevation  of 
Fig.  43,  only  two  of  these  elements  (viz.,  A  B  and  A  C^  can 
be  shown  in  their  true  length — all  the  others  are  foreshort- 
ened. This  must  be  borne  in  mind  by  the  student,  for  it 
is  evident  that  only  on  such  lines  as  are  shown  in  their  true 
length  can  measurements  be  obtained  for  any  points  of 
intersection.  Those  points  that  are  located  on  foreshort- 
ened elements  are  determined  in  a  particular  way,  as  will 
appear  during  the  construction  of  the  following  problem. 

The  plan  and  front  and  side  elevations  and  the  full  view 
of  the  section  for  each  problem  and  case  are  to  be  drawn 
on  this  plate  in  the  same  corresponding  relation  as  on  the 
preceding  plate. 


probIjEM  so 

88.     To  project  sectional  views  of  a  cone. 

The  three  cases  of  this  problem  should  be  carefully  stud- 
ied by  the  student,  for  on  the  application  of  the  principles 
here  shown  depend  nearly  all  operations  of  pattern  drafting 
that  relate  to  so-called  flaring  work.  A  clear  conception  of 
the  methods  used  will  be  found  indispensable  to  the  drafts- 
man that  desires  to  become  proficient  in  his  work.  An 
effort  should  be  made  on  the  part  of  the  student  to  trace 
each  operation  to  its  fundamental  principle,  when  it  will  be 
discovered  that  the  drawing  practically  resolves  itself  into 
a  continuation  of  problems  relating  to  the  true  lengths  of 
foreshortened  lines.  Since  such  problems  have  occupied 
his  attention  during  the  drawing  of  the  earlier  plates,  he 
should  have  no  difficulty  in  following  the  constructions  here 
given. 

Case  I. —  Wlicn  the  cutting-  plane  is  oblique  to  the  base 
and  intersects  all  the  elements  of  the  cone. 

The  position  of  the  cone,  its  dimensions,  and  the  angle  of 
the  cutting  plane  are  shown  in  Fig.  42. 

Construction. — Draw  a  right  plan  and  elevation  and 
represent  the  cutting  plane  ni  n  (see  plate)  by  a  line  drawn 


84 


PRACTICAL    PROJECTIOX 


15 


at  an  angle  of  45°  with  B  C,  cutting  A  B  h  inch  from  B. 
Divide  the  circle  that  represents  the  outline  of  the  base  of 
the  cone  into  any  convenient  number  of  equal  parts  (in  this 
case  16).  Draw  lines  from  these  points  {b,  1,  2,  3,  etc.)  to 
the  center;  next,  project  these  lines — or  elements,  as  they 
are  called — to  the  elevation.  Project  the  intersections  of 
the  elements  in  the  elevation  with  the  cutting  plane  ;;/ ;/ 
to  the  corresponding  elements  in  the  plan  and  trace  a  curve 
through  the  points  thus  obtained. 

Attention  is  called  to  the  fact  that,  although  the  distances 
between  the  points  found  in   this  manner  are  unequal,  yet 


Fig.  43  Fig.  « 

the  foreshortened  view  of  the  sectional  surface  thus  shown 
in  the  plan  is  a  true  ellipse,  as  is  also  the  full  view  of  the 
section  next  to  be  drawn.  Project  a  side  elevation  to  the 
right  (see  Art.  61),  also  showing  a  foreshortened  view  of 
the  section — an  ellipse  of  a  different  curvature. 

Case  II. —  When  the  eutting  plane  is  parallel  to  one  of  the 
elements,  the  section  being  in  this  ease  a  i>ai-abola. 


§  15  PRACTICAL    PROJECTION  86 

Explanation'. — Fig.  43  is  a  right  plan  .  and  elevation, 
giving  the  dimensions  of  the  cone.  The  cutting  plane  mn 
is  in  this  case  parallel  to  the  side  A  B  of  the  cone  and 
cuts  B  C  ^  inch  from  /)'.  The  method  of  drawing  these  pro- 
jections is  precisely  similar  to  that  in  the  preceding  case, 
and  since  corresponding  points  are  similarly  designated  in 
the  different  views  shown  on  the  plate,  the  student  should 
experience  no  difficulty  in  completing  the  drawing. 

Case  III. —  Wlicii  t/ic  cuttiiie;  plane  is  perpendicular  to  tJic 
basc  of  the  cone,  the  section  being  an  hyperbola. 

Explanation. — To  produce  that  section  of  the  cone 
known  as  the  hyperbola,  the  cutting  plane  may  form  with 
the  base  any  angle  included  between  a  right  angle  and  the 
angle  formed  by  an  element  with  the  base  (as  the  angle  A  B  C, 
Fig.  44).  The  dimensions  and  position  of  the  cone  and 
the  cutting  plane  are  shown  in  Fig.  44,  and  since  the 
method  of  projection  is  the  same  as  in  the  two  preceding 
cases,  the  student  may  complete  the  views  without  further 
instruction.  In  this  case,  the  projection  of  a  separate  full 
view  may  be  omitted,  the  latter  being  shown  in  the  side 
elevation. 


PROBLEM   31 

80.     To  project  sectional  vicAvs  of  a  scalene  cone. 

Explanation. — This  solid,  whii:li  is  of  varied  form  and 
of  frequent  occurrence  in  the  metal  trades,  is  an  irregular 
geometrical  figure.  It  is  a  cone  whose  axis  is  inclined  to  its 
circular  base.  All  the  elements  of  a  regular  cone  are  of  equal 
length,  Init  the  elements  of  a  scalene  cone  are  necessarily 
of  variable  length,  for,  since  its  axis  is  inclined  towards  a 
portion  of  the  base,  the  elements  in  that  part  of  the  surface 
must  be  shorter.  It  is  to  be  noted  in  this  case  that  the  axis 
of  the  cone  shown  in  Fig.  45  does  not  pass  through  the 
center  of  the  circle  that  represents  the  base  of  the  solid. 

Construction. — To  reproduce  this  drawing  on  the  plate, 
draw    first    the    horizontal   line   ba  in  the    plan    3§    inches 


86 


PRACTICAL    PROJECTION 


§  15 


long,  as  called  for  by  the  dimension  figures  in  Fig.  45 ;  next. 

describe  the  circle  boc 
2i  inches  in  diameter 
from  a  center  located 
on  the  line  ba,  its  circum- 
ference passing  through 
the  point  b.  Project  the 
elevation  according  to 
the  dimensions  given  in 
Fig.  45.  The  axial  line  is 
next  drawn ;  bisect  the 
angle  B  A  C  and  draw  the 
bisector  A  0\  represent 
the  cutting  plane  w  ;;  by 
a  line  drawn  perpendicular 
to  A  O  and  cutting  A  C 
f  inch  from  C. 

The  method  of  projec- 
tion for  the  various  views 
of  this  solid  is  the  same 
is,  divide  the  outline  of  the 
convenient  number  of  spaces 
(12  in  this  problem),  and  from  points  thus  located  draw 
the  elements  to  the  apex  a  (see  plate).  Next,  project 
these  elements  to  the  elevation  and  finalh''  project  their 
intersections  with  the  cutting  plane  to  the  different  views,  as 
shown  in  the  drawings  on  the  plate.  Note  that  the  full 
view  of  the  section,  which  in  this  problem  is  taken  at  right 
angles  to  the  axis  of  the  solid,  is  an  ellipse.  If  the  cone 
were  in  an  upright  position — that  is,  with  its  base  at  right 
angles  to  the  axis  (as  that  part  of  the  cone  above  the  cut- 
ting plane  in  Fig.  45) — the  solid  would  be  termed  an 
"elliptical"  cone.  This,  strictly,  is  not  a  geometrical 
solid,  but  its  characteristics  in  projection  drawing  are  some- 
what similar  to  the  regular  cone,  although  the  elements  in 
each  quarter  of  the  base  are  alwaj-s  of  unequal  length. 

The  student  should  now  be  able  to  project   any   sections 
that  mav  be  desired. 


Fig.  45 
as   in    former   cases;    that 
base    in    the    plan    into    a 


JUNEa5JS93. 


Copyright,  1899,  by  THE  Co 
All  riEfht 


TIDNS-I. 


I  I  II 1 1  I  /1 1/  /  /. 

'■  ■  \\  \\  III!  If 


•-'''  "■'— -N^ 


P/fOBLEM  Sa. 
Ccrse  3. 


C  B 

PROBLEM  £3. 


ERY  Engineer  Company. 
served. 


JO^Af  3M/  T/i.  CL  ^SSy/^  4529. 


§  15  PRACTICAL    PROJECTION  87 

DRAWII^^G    PIRATE,   TITLE:    INTERSECTIONS    I 

90,  The  IVIiter  Line. — To  represent  properly  the  inter- 
sections of  the  surfaces  of  solids — or  to  "draw  the  miter 
line,"  as  it  is  commonly  called — is  the  final  process  of  pro- 
jection. It  has  already  been  remarked  that  plane  surfaces 
intersect  in  a  line  ;  the  representation  of  the  intersection 
of  plane  surfaces  is  therefore  a  very  simple  process, 
the  draftsman  merely  having  to  define  each  surface  by 
the  application  of  the  regular  projection  methods  already 
explained.  The  intersection  of  curved  surfaces  is  appar- 
ently more  complicated,  but  only  because  it  is  necessary  to 
locate  a  greater  number  of  points  than  are  required  for  the 
intersection  of  plane  surfaces.  The  location  of  points  for 
the  representation  of  the  intersection  of  curved  surfaces  is 
done  in  a  manner  somewhat  similar  to  that  already  shown 
in  connection  with  the  projection  of  plane  surfaces  having 
curved  outlines.  There  is,  however,  this  important  differ- 
ence to  be  observed :  in  the  case  of  the  surfaces  mentioned, 
their  projection  is  accomplished  by  means  of  points  located 
on  their  outlines;  while  in  the  case  of  the  intersection  of 
curved  surfaces,  it  is  necessary  to  locate  lines  in  such  posi- 
tions on  each  surface  that  they  will  lie  in  the  same  plane, 
although  drawn  on  different  surfaces.  It  is  possible  to  locate 
a  number  of  these  lines  in  such  positions  on  the  drawing  that 
through  their  points  of  intersection  a  curve  may  be  traced 
that  will   be  the  correct  line  of  intersection  of  the  surfaces. 

91.  Relation  of  the  Miter  Line  to  the  Pattern. — No 

drawing  of  an  object  in  which  intersected  solids  are  repre- 
sented is  complete  unless  the  line  of  intersection  is  accu- 
rately produced.  This  is  a  very  important  part  of  the 
drawing,  and  the  correct  "fit"  of  the  pattern  in  work  of 
this  class  often  depends  entirely  on  the  accuracy  with  which 
the  line  of  intersection  is  drawn.  In  fact,  a  development,  or 
pattern,  cannot  be  made  until  the  drawing  is  complete  in 
this  particular.  The  principles  governing  the  use  of  these 
lines  are  clearly  shown  in  the  explanation  accompanying 
the  problems,  which  the  student  should  study  carefully;  for 


88 


PRACTICAL    PROTECTION 


15 


if  he  thoroughly  comprehends  the  principles  governing 
their  use  and  exercises  due  care  to  see  that  lines  are  drawn 
from  the  same  corresponding  points  in  each  view,  he  will 
have  no  difficulty  in  producing  the  correct  lines  of  intersec- 
tion for  the  surfaces  of  the  solids  represented  on  these 
plates,  or.  in  fact,  for  the  surfaces  of  any  solid.  The  prob- 
lems for  this  plate  consist  of  the  projection  of  intersecting 
solids  having  plane  surfaces.  The  solids  are  shown  in  per- 
spective in  the  illustrations  accompanying  each  problem, 
and  reference  to  the  projections  on  the  plate  will  be  suffi- 
cient to  show  the  method  of  finding  the  lines  of  intersection. 


PROBLEM  35 

92.     To  project   views  of  intersecting  prisms. 

When  the  plane  or  curved  surfaces  of  any  solid  so  inter- 
sect as  to  present  one  continuous  surface — that  is,  so  that 
the  surfaces  meet  "edge  and  edge"  in  the  same  plane — no 
line  of  intersection  is  necessary,  since  such  surfaces  are 
relatively  in  (and  a  part  of)  the  same  plane.  When,  how- 
ever, the  surfaces  of  one  solid  intersect  a  central  portion  of 
the  surface,  or  surfaces,  of  another  solid,  it  is  necessary  that 
the  line  of  intersection — that  is,  the  boundary  lines  of  the  sur- 
faces of  the  intersecting  solid — should  be  accurately  drawn. 

Case  I. —  When  the  axes  of  the  prisms  intersect  at  an 
angle  of  90°. 

ExpLAXATiox. — The  solids  for  the  projeciion  of  this  prob- 
lem are  shown  in  perspective  in  Fig.  -16,  which  represents 
also  their  position  in  the  intersection  of 
Case  I.     The  figure  consists  of  an   up- 
right shaft  in  the  form  of  a  quadrangu- 
lar prism  with   a  horizontal  octagonal 
^  prism  intersecting,   or    **mitering, "    at 
right  angles  along  the  axial  lines  of  the 
I    two   solids.      The    projection    and    ar- 
rangement of  the  solids  is  seen  in  the 
drawings  for  this  problem  on  the  plate. 
It   will    be   noticed    that    the    octagons 


§  15  PRACTICAL    PROJECTION  89 

drawn  in  dotted  lines  in  the  rigure  do  not  form  a  part  of  the 
finished  drawing,  but  are  thus  drawn  to  facilitate  the  pro- 
jection, as  the  edges  of  the  solids  may  thus  be  determined 
before  the  side  view  is  drawn.  It  is  often  convenient  to 
place  portions  of  views  temporarily  on  the  drawing  in  this 
manner,  as  much  labor  is  thereby  saved.  The  plan  is  com- 
pleted first  and  the  remainder  of  the  drawing  finished  in  the 
usual  way,  as  may  be  seen  from  the  reduced  copy  of  the 
plate.  It  is  thus  shown  that  the  correct  line  of  intersection 
is  found  by  simple  methods  of  projection,  the  position  of 
the  points  in  the  line  being  determined  by  projection  from 
the  different  views. 

Construction. — Draw  the  plan  ABC  D  in  its  proper 
position,  as  shown  on  the  plate,  and  next  construct  the 
elevation  A  A'  C  L\  thus  completing  the  projection  of  the 
quadrangular  prism  as  though  that  solid  alone  were  to  be 
represented.  Draw  a  horizontal  axial  line  (not  shown  on 
the  plate)  for  the  octagonal  prism  through  both  views,  and 
at  the  left  of  the  views  thus  drawn  construct  a  full  view  of 
the  end  of  the  octagonal  prism,  as  shown  by  the  octagons 
in  dotted  lines  on  the  plate.  Next,  draw  the  lines  that  rep- 
resent the  edges  of  the  octagonal  prism  in  both  views. 
From  the  points  of  intersection  of  such  lines  in  the  plan 
with  the  edges  A  B  and  A  D  of  the  quadrangular  prism, 
draw  primary  projectors  to  corresponding  lines  in  the  eleva- 
tion. Draw  connecting  lines  through  the  points  of  inter- 
section thus  determined  ifi  the  elevation,  as  shown  at  a'  b'  c'  d' 
on  the  plate.  This  completes  the  projection  of  that  view; 
the  side  elevation  may  then  be  projected  by  means  of  sec- 
ondary projectors,  as  in  former  problems. 

This  plate  is  divided  in  the  same  manner  as  the  preceding 
plate,  and  the  projections  for  this  case  are  drawn  in  the 
upper  left-hand  space. 

Case  II. —  WlicH  the  axes  intersect  at  an  oblique  angle. 

This  projection  is  completed  as  shown  on  the  j)Iate.  In 
this,  as  in   the  preceding  case,  the  plan  is  first  drawn;  but 


90  PRACTICAL     PROJECTION  §  15 

before  it  can  be  completed,  a  portion  of  the  elevation  has  to 
be  drawn  in  order  to  determine  the  outline  of  the  octagonal 
surface  in  the  plan  (Problem  6). 

CoxsTRUCTiox. — First  draw  the  outline  A  BCD  in  the 
plan  and  then  construct  the  octagon  shown  in  dotted  lines 
at  {a),  from  the  points  of  which  draw  horizontal  lines  to  the 
plan  of  the  quadrangular  prism,  in  the  manner  shown. 
Next  draw  the  elevation  of  the  quadrangular  prism  and 
locate  the  point  x  midway  on  the  line  A  A' :  the  angle  of 
inclination  of  the  octagonal  prism  is  in  this  case  15',  and  a 
line  at  that  angle  is  then  to  be  drawn  through  j:  for  the 
axial  line  of  that  solid.  Construct  the  full  view  of  the  end 
of  the  octagonal  prism  at  (d),  and  from  the  points  t\  f.  g,  etc. 
in  that  view  draw  lines  of  indefinite  length  towards  the 
right ;  intersect  these  lines  at  a\  b\  c\  and  d'  in  the  elevation 
by  vertical  primary  projectors  drawn  from  corresponding 
points  in  the  plan,  thus  establishing  the  line  of  intersection 
in  the  elevation.  The  plan  is  then  completed  by  drawing 
primary  projectors  vertically  downwards  from  the  edge  view 
of  the  end  of  the  octagonal  solid  in  the  elevation  and 
tracing  the  outline  of  the  inclined  surface  thus  designated, 
as  in  Problem  6.  The  side  elevation  is  next  projected  by 
the  arc  method  of  secondary  projectors,  as  heretofore. 
Xote  that  the  intersection  of  the  upper  portion  of  the 
octagonal  surface  is  to  be  represented  in  that  view  by 
dotted  lines. 

Case  III. —  Wlien  the  axes  are  at  an  oblique  angle,  but  do 
not  intersect. 

ExpLAXATiox. — The  position  of  the  solids  is  shown  in  the 
drawings  on  the  plate,  and  the  projections  do  not  differ 
materially  from  those  of  the  two  preceding  cases.  The 
octagonal  solid  is  in  this  case  inclined  at  an  angle  of  30°, 
and  its  axial  line  m  //,  as  shown  in  the  plan,  is  drawn  ^  inch 
below  the  center  of  the  quadrangular  prism.  The  order  of 
procedure  is  the  same  as  that  given  for  the  drawing  of 
Case   II  and  will  present  no   difficulties   to   the    attentive 


§  15  PRACTICAL    PROJECTION  91 

student.  It  is  necessary,  however,  to  use  extreme  care  in 
these  projections,  in  order  that  the  position  of  points  in  the 
different  views  may  be  located  i)i  the  same  corresponding 
position  zvith  regard  to  one  anotJier.  * 


PROBLKM    33 

93.     To   project  vie^vs  of  a  prism  intei-sectecl   by  a 
cylinder. 

Explanation. — Fig.  47  is  a  perspective  view  of  an  octag- 
onal prism  intersected  by  a  cylinder  at  an  oblique  angle,  the 
axes  of  the  solids  not  intersect- 
ing. The  arrangement  of  the 
views,  the  dimensions,  and  the 
angle  of  inclination  are  shown  on 
the  plate.  The  projection  of  this 
problem  is  very  similar  to  the  last 
case  of  Problem  "I'l.  Note  that 
the  position  of  the  two  solids  is 
such  that  the  axis  of  the  cylinder 
intersects  the  edge  D  of  the 
prism.   The  circles  that  represent  ^'"^-  ■*" 

the  end  surface  of  the  cylinder  in  each  view  are  divided  into 
a  convenient  number  of  equal  spaces,  and  from  the  points 
thus  located  lines  parallel  to  the  axis  of  the  cylinder  are 
drawn  on  its  surface.  These  lines  are  then  projected 
in  the  same  way  as  the  lines  that  represented  the  edges 
of  the  octagonal  prism  in  Problem  2'-i.  Since  the  surface  of 
the  cylinder  is  a  curved  surface,  the  line  drawn  through 
the  points  of  intersection  of  the  two  solids  will  be  a  curved 
line. 


*  Too  much  stress  cannot  be  laid  on  this  important  statement ;  as  the 
student  progresses  with  the  projection  of  the  views,  he  will  see  tlie  im- 
portance of  this  matter.  The  work  must  be  done  slowly,  keeping  the 
pencil  points'well  sharpened;  the  position  of  the  points  in  the  drawing 
may  then  be  accurately  delermined,  if  the  views  are  constantly  com- 
pared. If  this  is  done,  there  will  be  no  difficulty  attached  to  any  of  the 
problems  to  follow  in  this  stection. 


92  PRACTICAL    PROJECTION  §  15 

/^  94.     General  Instruction  Relating  to  Intersections. 

When  points  are  located  on  the  full  view  of  a  surface 
in  a  plan  and  elevation,  as  on  the  circles  at  (a)  and  (d)  in 
the  drawings  on  the  plate  for  the  last  problem,  care  must  be 
taken  that  the  distinction  between  the  different  views  is 
maintained;  thus,  the  point  x,  at  (a)  and  in  the  plan,  is 
located  at  .v'  at  (d)  and  in  the  elevation,  both  positions  on 
the  drawing  representing  the  same  position  on  the  solid. 
Any  other  points  thus  located  on  an  outline  will  be 
changed  in  a  corresponding  way  with  relation  to  one 
another.  To  project  intersections  of  solids,  all  of  whose 
surfaces  are  bounded  by  parallel  lines  or  on  whose  sur- 
faces parallel  lines  may  be  drawn  that  will  also  be  parallel 
'to  the  axis  of  the  solid,  it  is  necessary  first  to  draw  a 
view  that  will  show  the  intersectr^:/  surface,  or  surfaces, 
in  one  line — that  is,  "on  edge."  Such  views  have  been 
drawn  in  the  plans  of  the  preceding  problems.  The  lines 
of  the  intersect/';/^  solid  are  then  represented  in  this  view, 
and  the  points  of  their  intersection  with  the  upright 
surfaces  are  projected  to  the  elevation  in  the  manner 
described. 

It  will  be  seen  that,  if  all  the  surfaces  of  any  intersecting 
solids  are  plane,  their  edges  or  outlines  alone  will  suffice  for 
finding  the  lines  of  intersection.  If  the  surfaces  are  curved, 
it  is  merely  necessary  to  locate  a  number  of  points  on  the 
outline  of  the  full  view,  through  which  to  draw  parallel  lines 
similar  to  those  drawn  on  the  cylinder  in  Problem  23.  This 
practically  changes,  or  reduces,  the  cylinder  to  a  solid 
bounded  by  a  number  of  plane  surfaces.  In  the  case  of 
Problem  23,  the  cylinder  has  really  been  treated  as  though 
it  were  a  prism  having  a  number  of  sides  equal  to  the  num- 
ber of  spaces  into  which  the  circles  were  divided.  This 
would  actually  have  been  the  case  had  straight  lines  been 
drawn  between  the  points  thus  located  on  the  circles 
at  (a)  and  (d).  Had  this  been  done,  the  line  of  intersec- 
tion would  also  have  been  represented  by  a  series  of  short 
lines  drawn  between  the  points  located  by  projection 
methods. 


INTER5E 


i^r 


^'  \r 


M  !    niii 


PROBLE/^  a4: 


\ 


\ 


~~~^^^^-^\         \  \ 


fW)BLEM26 


JUNE  25.1893. 


Copyright,  l5S9,  by  THE  CoiJ 


All  rights! 


:TiDN5-n. 


PROBLEM  as. 


t?   d  f  hy 


Pf?OBLEM  £7. 


<v  Engineer  Company, 
srved. 


JOH/^  SMJTH^  CLASS  /^e4S29. 


15 


PRACTICAL    PROJECTION 


93 


DRAWING    PLATE,  TITLE:    INTERSECTIONS    II 

95.  This  plate  is  dividetl  in  the  same  manner  as  the  pre- 
ceding plate,  and  the  problems  occupy  the  same  relative 
positions. 

96.  Iiitei'sections  of  Cylinders.  —  Tlic  intersection 
lines  of  cylinders  equal  in  diameter  a)id  intersecting  one 
another  in  the  same  plane — that  is,  so  that  their  axes  also 
intersect — are  always  represented  by  straight  lines  iti  a  vieiu 
that  shotvs  the  axes  in  their  true  length.  This  is  the  case  in 
the  projections  that  are  shown  in   Fig.  48,  which  represents 


Fig.  48 


objects  that  are  familiar  to  all  sheet-metal  workers;  namely, 
pipe  angles  {a),  elbows  {b),  Y's  {c),  and  T's  {d).  The  same 
is  also  true  of  similar  moldings  "  mitered  "  in  a  plane  in 
which  the  true  length  of  all  members  of  such  moldings  may 
be  shown.  The  line  of  intersection  may  always  be  found  in 
such  cases  by  bisecting  the  angle  made  by  the  pipe  or  mold- 
ings. Fig.  48  also  illustrates  an  example  of  line  shading 
often  used  by  draftsmen  to  designate  cylindrical  surfaces  on 


M.  E.    V.-iS 


94 


PRACTICAL    PROJECTION 


§15 


working  drawings.  When  cylinders  of  different  diameters 
intersect  (or  cylinders  of  the  same  diameter  whose  axes  do 
not  intersect),  the  lines  of  intersection  in  any  view  are 
curved  lines,  and  are  found  in  a  manner  similar  to  the  pro- 
jection in  the  last  problem.  The  lines,  however,  that  are 
drawn  on  any  one  cylinder  must  be  projected  to  the  inter- 
secting cylinder,  for  it  is  at  the  intersection  of  lines  thus 
drawn  that  points  are  established  through  which  the  curve  of 
intersection  may  be  traced.    This  is  illustrated  in  Problem  24:. 


PROBLEM    24 

9T.     To  project  A'ievrs  of  intersecting   cylinders  of 
unequal  diameters. 

Explanation. — Fig.  49  is  a  perspective  view  of  a  branch  Y 
of  occasional  occurrence  in  blowpipe  work.  The  arrange- 
ment of  the  views  and  the  method  of  projec- 
tion are  shown  on  the  plate,  and  since  it  is 
similar  to  those  of  drawings  that  have  been 
made  on  the  preceding  plate,  no  definite 
instructions  are  necessary.  The  diameter 
of  the  larger  cylinder  is  2  inches  and  that  of 
the  smaller  1  inch,  while  the  angle  of  inter- 
section shown  in  the  elevation  is  45^.  The 
position  of  the  two  cylinders  is  such  that 
the  outline  r  s  oi  the  smaller  cylinder  in  the 
Fig.  49  plan  is  tangent  to  the  circle  that  represents 

the  large  cylinder.  Note  the  position  of  points  on  the  circle 
that  represents  the  end  surface  of  the  smaller  cylinder  in 
the  plan,  at  a,  b,  and  r,  and  their  corresponding  location 
at  a\  b',  c',  etc.  in  the  elevation  (Art.  94).  Also  observe 
that  certain  points  on  lines  drawn  on  the  surface  of  the 
smaller  cylinder,  as  the  lines  ;;/;/,  p  q,  and  rs'xn  the  plan,  are 
projected  to  the  front  elevation,  where  they  are  represented 
as  lines  at  ;/'  ;/",  q'  q",  and  s'  s" .  These  corresponding  lines 
are  in  the  same  plane  and  are  the  lines  drawn  on  the  surface 
of  the  larger  cylinder,  as  above  mentioned.  Through  the 
points  of  their  intersection  with  lines  drawn  from  a'  b\  and  c' 


§  15  PRACTICAL    PROJECTION  96 

in  the  elevation,  the  curved  line  of  intersection  of  the  two 
cylinders  is  to  be  traced,  thus  completing  the  problem. 
Project  the  side  elevation  as  heretofore. 


PROBLEM    35 

98.  To  project  views  of  a  cone  Intersected  by  a 
cylinder,  the  axes  of  tlie  two  solids  being  i^arallel. 

Explanation. — Fig.  50  is  a  perspective  view  of  a  steam- 
exhaust  head,  a  modification  of  which  is  in  common  use. 
The  drawing  on  the  plate  is  a  Com-  ||,||,,..... 

plete  projection  of  the  same,  in  which  '^^^~~~_  1^^^!!^ 
the  proportion  of  the  cylinder  C  is  in-  ^HHH||'  b^^^w 
creased,  in  order  to  show  the  method      ^  ^^^^f 

of    projection    to    better    advantage.         ^^^fc         ^^ 
Observe  that  the  position.of  the  object  ^^B       W 

is  reversed,  as  the  drawing  is  thereby  ^^g      W 

facilitated.       The    elevations    on    the  ^p    a 

plate  show  that  the  lines  of  intersec-  HIOHJ" 

tion  of  the  two  cylinders  A  and  B  with  IBILJ' 

the  cone  are  represented   by  straight  ^'^-  ^ 

lines;  this  is  always  the  case  when  a  cylinder  whose  diameter 
is  equal  to  a  section  of  the  cone  intersects  in  this  manner. 

Construction. — First,  construct  the  elevation  of  the  cone 
in  its  proper  place  on  the  drawing  and  in  accordance  with 
the  dimensions  given  on  the  plate.  The  cylinders  A  and  B 
are  then  drawn  as  shown.  Next,  draw  the  plan  and  repre- 
sent the  outline  of  the  cylinder  C  in  that  view  by  a  circle 
of  the  diameter  indicated  on  the  plate.  Divide  this  out- 
line into  a  convenient  number  of  equal  spaces  (in  this 
case  8),  as  shown  at  a.  d,  c,  etc.,  and  through  each  of  these 
points  draw  elements  of  the  cone,  as  O  />,  O  (/,  etc.  Project 
these  elements  to  the  elevation  and  locate  thereon  by  pro- 
jection methods  the  position  of  the  points  of  intersec- 
tion a',  h\  g\  etc.  Complete  the  elevation  by  drawing  the 
outline  of  the  cylinder  C  in  that  view  and  tracing  the  line 
of  intersection  of  the  two  solids  through  the  points  a\  h\  g' 
etc.     Project  the  side  elevation  by  the  usual  methods. 


96 


PRACTICAL    PROJECTION 


§15 


Fig.  51 


PROBLEM    26 

99.     To  project  views  of  iutersecting  cones. 

Fig.  51  is  a  perspective  view  of  an  object  that  illustrates 
this  problem.  The  cones  are  shown 
in  a  somewhat  more  convenient 
proportion  for  this  problem  in  the 
projections  on  the  plate.  The  con- 
struction of  this  problem  must  be 
followed  very  carefully,  as  a  number 
of  the  operations  are  necessarily 
made  over  one  another  on  the  draw- 
ing and  the  student  must  be  careful 
to  distinguish  each  process. 

CoxsTRUCTiox. — Describe  a  circle 
•2|-  inches  in  diameter  in  the  plan 
to  represent  the  lower  base  of  the 
larger,  or  intersected,  cone  in   that 

view;  and  from  the  same  center  describe  a  circle  f  inch  in 

diameter,  to    represent  the 

upper    base.      Project    the 

front  elevation  of  this  cone 

and     define      the     frustum 

2^  inches  high,  as  shown  on 

the    plate,    producing    the 

outlines  until  they  meet  in 

the  vertex.      Next,  through 

the  point  F,  Fig.  S'^jdraw  the 

axial    line    of    the    smaller, 

or    intersecting,    cone    (the 

line    A  B,    Fig.    52)  at   an 

angle  of  45°  with  the  base 

of  the    larger  cone;    locate 

the  point  A   3  inches  from  „ 

F,    and,     after     fixing    the 

point    C   1    inch    from    F, 

draw     C  D     perpendicular 

to  A  B.      Draw  the  outline 

of    the    upper    base    of    the  fig.  53 


§  15 


PRACTICAL    PROJECTION 


97 


smaller  cone  in  the  elevation  parallel  to  C  D  and  f  inch 
from  A.  Draw  tlie  full  view  of  the  base  of  the  smaller  cone 
zX  J  B  //;  divide  this  outline  into  a  convenient  number  of 
equal  parts,  as  at  J,  a,  B,c,  etc.,  and  project  these  points 
to  the  base  C  D.  Draw  the  elements  of  the  smaller 
cone,  as  shown  in  Fig.  5"2,  and  produce  them  until  they 
intersect  the  base  of  the  large  cone  at  /f,  /%  and  6^.* 
In  the  plan  of  this  drawing,  a  series  of  sections  of  both  cones 
are  now  drawn  as  the  sectional  curves  would  appear  if  each 
of  the  elements  of  the  smaller  cone  shown  in  the  elevation 
were  considered  as  a  cutting  plane,  as  in  Problem  20.  The 
sections  of  the  smaller  cone  will  in  each  case  be  a  triangle 
(Art.  86),  while  the  sections  of  the  larger  cone  will  be  ellip- 
tical, parabolic,  or  hyper-  .A 
bolic  curves,  as  the  case  may 
be.  The  point  of  intersec- 
tion with  the  side  of  each 
triangle  and  its  correspond- 
ing sectional  curve  of  the 
larger  cone  is  then  projected 
to  the  elevation  and  the  line 
of  intersection  of  the  two 
cones  traced  through  these 
points.  The  sectional  tri- 
angles of  the  smaller  cone 
are  shown  in  the  plan  of 
Fig.  52.  Draw  vertical 
primary  projectors  to  the  '^•^^—^ 
plan  from  points  a' ,  B\ 
and  t',  and  on  these  pro- 
jectors set  off  distances 
from  the  horizontal  center 
line  of  the  plan  similar  to 
the  distances  from  H /  '\n  the  full  view  of  the  base;  that  is, 
make  x'  c"  equal  x  c,  etc. 

*  In  this  case  eight  elements  are  repre.sented,  since  it  is  not  desirable 
to-  complicate  the  drawing  by  using  more,  although  in  practical  work 
it  will  be  found  necessary  to  use  a  larger  number  of  points  in  order 
that  the  line  of  intersection  may  be  more  accurately  traced. 


Fig.  53 


98 


PRACTICAL    PROJECTION 


15 


It  is  not  necessary  to  develop  the  sectional  curves  of  the 
larger  cone  in  their  entire  length,  since  all  that  is  required 
is  to  find  a  point  on  each  curve  that  is  at  the  intersection  of 
the  triangular  sections  of  the  smaller  cone.  This  is  better 
illustrated  in  Fig.  53,  in  which  is  shown  the  projection  of 
the  point  o  in  the  line  of  intersection.  A  study  of  this 
figure  will  show  to  the  student  that  the  operations  are  simi- 
lar to  those  of  Problem  "20;  the  process  in  the  case  of  each 
point  is  merely  a  repetition  of  that  here  indicated  and  need 
not  be  further  explained.  The  extreme  upper  and  lower 
points  J'  and  z.  Fig.  5'2,  are  the  points  of  intersection  of  the 
central  section  and  may  be  projected  directly  to  the  plan 
from  the  elevation,  since  the  outline  of  the  figure  in  the 
elevation  is  really  a  section  on  the  line  vi  Ji  in  the  plan  of 
Fig.  5'2.  Fig.  oi  shows  the  three  sections  produced  by  the 
above  method;  that  is,  the  irregular  curve  p  q  r.  Fig.   54, 

is  a  section  of  the  larger 
cone  produced  by  the  in- 
tersection of  the  cutting 
plane  A  E\  s  s  and  t  t 
are  found  as  above  de- 
scribed, the  section  lines 
being  indicated  in  the 
figure  by  short  cross- 
hatching.  Project  the  side 
elevation  as  in  former 
problems. 

The  student  may  at  the 
completion  of  the  drawing 
on  the  plate  erase  all  the 
construction  lines  except 
those  projectors  shown  on 
the  reduced  copy  of  the 
plate,  those  lines  only  be- 
ing inked  in  that  are  neces- 
sary to  show  the  outlines  of 
the  figure  and  the  line  of 
intersection  in  each  view,  as  well  as  the  outer  projectors. 


§15 


PRACTICAL    PROJECTION 


99 


PROBLEM    27 

100.  To  i>roject  views  of  a  sphere  intei*sected  by 
a  cylinder. 

When  the  position  of  these  two  solids  is  such  that  the  axis 
of  the  cylinder  passes  through  the  center  of  the  sphere,  as 
shown  in  perspective  in  Fig.  55,  the  line  of  intersection 
is  shown  in  a  right  elevation  as  a  straight  line.  In  the  case 
of  the  solid  shown  in  perspective  in  Fig.  56,  the  axis  of  the 
cylinder  does  not  pass  through  the  center  of  the  sphere,  and 


Fig.  55 


Fig.  56 


the  line  of  intersection  is  an  irregular  curve,  which  may  be 
found  by  the  following  method.  vSince  the  construction  of 
this  problem  involves  lines  that  must  necessarily  be  drawn 
closely  together  in  the  small  scale  adopted  for  these  drawings 
on  the  plate,  a  proportion  is  selected  that  does  not  admit  of 
the  entire  figure  being  shown  in  the  elevations;  these  pro- 
jections are  therefore  finished  by  broken  lines,  as  indicated 
on  the  plate. 

Construction. — Draw  the  plan  first  and  represent  the 
view  of  the  sphere  by  a  circle  SI  inches  in  diameter.  Draw  a 
vertical  diameter  and  from  a  point  midway  on  the  radius  ad 
describe  a  circle  If  inches  in  diameter,  to  represent  the  end 
view  of  the  cylinder,  on  the  outline  of  which  locate  a  number 
of  points  by  spacing  with  the  dividers,  as  at  1,  2,  3,  4,  etc. 
Through  these  points,  in  the  manner  shown  on  the  plate, 
draw  vertical  lines,  a.s  ad,  c  d,  ef,  gh,  and  i j\  each  of  which 
will  now  represent  a  cutting  plane.      Sections  of  the  sphere 


100  PRACTICAL     PROJECTION  §  15 

and  cylinder  on  these  lines  are  now  to  be  produced  in  the 
side  elevation.  These  sections,  as  already  stated,  are  circles 
and  parallelograms,  respectively,  the  diameter  of  the  circles 
being  ascertained  from  the  view  in  which  the  cutting  plane 
is  shown  on  edge,  as  in  the  plan.  In  this  problem,  the  side 
elevation  is  next  projected.  Describe  arcs  representing  the 
sections  in  the  side  elevation,  using  the  radii  ab,  c  d,  f^  f.gJi^ 
and  i j\  the  radius  a  b  being  the  great  circle  of  the  sphere. 
By  the  aid  of  primary  projectors,  project  the  points  from 
the  cylinder  in  the  plan  to  the  side  elevation,  intersecting 
corresponding  arcs  in  the  side  elevation.  Trace  the  irregular 
curve  shown  in  that  view  on  the  plate  through  these  points. 
Next  project  the  front  elevation,  thus  completing  the 
problem. 

101,  Recapitulation.  —  The  problems  of  this  plate 
have  afforded  the  student  an  opportunity  for  careful  study. 
Owing  to  the  number  of  points  necessary  to  be  found  in 
each  figure,  the  problerhs  may  have  had  the  appearance  of 
more  or  less  complication,  but  if  the  student,  as  previously 
cautioned,  will  carefully  locate  the  points  required,  one  at  a 
time,  not  hurrying  his  work  nor  trying  to  grasp  the  entire 
problem  at  once,  but  keeping  in  mind  the  different  principles 
in  the  order  presented,  and  by  referring,  if  necessary,  again 
and  again  to  the  primary  principles,  he  will  experience  no 
difficulty  in  making  the  drawings.  He  will  also  be  able 
intelligently  to  project  any  view  of  any  object;  in  other 
words,  he  will  be  able  to  make  any  working  drawing  what- 
ever, and.  in  addition  to  this,  be  able  to  read  and  understand 
any  working  drawing  he  may  be  called  on  to  examine. 


DBVELOPMENT    OF    SURFACES 


IKTROD  UC  TICK 

1.  Defliiitiou  of  a  Development. — A  development  is 
a  drawing  in  which  a  full  view  of  all  the  surfaces  of  a  solid 
is  represented.  Whenever  a  development  is  to  be  drawn 
(except  in  the  case  of  solids  of  very  simple  form),  a  pro- 
jection drawing  must  first  be  made.  This  projection  should 
show  the  solid  in  a  right  position.  Since  the  location  of  the 
various  points  in  a  development  is  dependent  on  their  corre- 
sponding position  in  the  projection  drawing,  the  importance 
of  the  projection  and  the  necessity  for  accuracy  in  its  con- 
struction are  thus  clearly  seen.  If  a  solid  is  bounded  entirely 
hy  plane  surfaces,  its  development  can  be  accomplished  by 
merely  projecting  their  full  views,  as  already  explained  in 
Practical  Projection. 

A  solid  is  said  to  be  developed  when  all  surfaces  compo- 
sing it  are  represented  on  one  plane  and  in  such  relation  to 
one  another  that,  if  formed  or  bent  up,  they  will  constitute 
a  solid  similar  to  the  one  represented  by  the  projection  draw- 
ing from  which  the  development  was  made.  Such  a  rep- 
resentation is  called  a  development,  or  a  pattern,  the 
process  of  laying  out  the  pattern  being  termed  developing 
the  surfaces  of  the  solid. 

3.  Relation  of  the  Surfaces  in  a  Pattern. — When 
it  is  desired  to  produce  a  pattern  requiring  a  combination 
of  several  surfaces  that  are  adjacent  in  a  solid,  such  surfaces 
must  be  drawn  in  the  same  relation  to  one  another  in  the 

§10 

For  notice  of  c<)pyri>rht,  see  page  immediately  followinvj  the  title  page. 


2  DEVELOPMENT  OF  SURFACES  §  16 

development.  The  surfaces  of  a  solid  when  thus  combined 
in  a  pattern,  or  development,  bear  the  same  relation  to  one 
another  that  they  would  if  they  were  considered  as  being 
unfolded  or  unrolled — the  same  relation  that  a  paper  wrap- 
per would  bear  to  the  package  from  which  it  had  been 
unfolded  or  unrolled.  The  paper  wrapper  is  not  always  an 
apt  illustration,  as  the  metal  worker  seldom  requires  several 
thicknesses  of  his  material.  In  the  case  of  the  familiar 
•'square  pan,"  however,  the  ends  are  folded  on  one  another 
in  precisely  the  same  way  as  in  the  pai>er  wrapf>er. 

It  will  be  seen  from  the  foregoing  that,  were  all  solids 
bounded  by  flat,  or  plane,  surfaces,  the  subject  of  develop- 
ments would  present  no  new  problems;  it  would  be  neces- 
sary merely  to  study  the  relation  of  surfaces  to  one  another, 
project  their  full  views,  and  carefully  redraw  them  in  the 
pattern  in  the  same  relative  position. 

3.  Projection  Method*  I"«^<1. — It  has  been  shown  in 
Practical  Projecv.  r.  ::..^:  a  single  surface  is  developed,  or,  as 
stated,  its  full  view  is  drawn,  by  a  modification  of  the  same 
methods  that  are  used  to  produce  the  different  views  of  that 
surface.  Many  of  the  operations  attendant  on  the  develop- 
ment of  solids  are  like  those  used  in  producing  full  views  of 
single  surfaces:  or,  if  not,  the  principles  involved  may  be 
traced  to  their  origin  in  other  methods  used  in  projection 
drawing. 

A  thorough  knowledge  of  projection  is  absolutely  neces- 
sary that  the  student  may  understand  the  operations  involved 
in  developing  the  surfaces  of  a  solid.  The  position  of  the 
several  points  located  in  a  drawing  and  their  corresponding 
location  in  an  imaginary  way  on  the  object  itself  must  be 
definitely  fixed  in  the  student's  mind.  Each  line  must  be 
determined  in  its  relation  to  the  other  lines  of  the  drawing 
and  its  ideal,  or  imaginary,  location  definitely  ascertained ; 
the  surfaces,  also,  must  be  treated  in  a  similar  way.  The 
student  must  picture  to  himself  the  completed  object  as  it 
will  appear  when  the  surfaces  laid  out  on  the  drawing  board 
in  the  development  are  formed  up  in  their  final  relation  to 


§  16  DEVELOPMENT  OF  SURFACES  3 

one  another.  This  imaginary  part  of  the  study  is  of  even 
greater  importance  in  the  case  of  developments  than  in  pro- 
jection drawing.  As  the  student  has  already  had  some  drill 
in  this  part  of  the  work,  the  subject  he  is  now  studying 
should  be  found  less  difficult  than  would  otherwise  be  the 
case.  In  projection  drawing,  the  surfaces  of  the  solid  are 
represented  as  being  in  their  proper  position ;  in  the  develop- 
ment, the  same  surfaces  are  represented  as  being  developed 
or  spread  out  on  the  surface  of  the  drawing  board. 


GrE:N^ERAL  classificatio:n^ 

4.  General  Classification  of  Solids. — An  accurate 
development  may  be  drawn  for  the  plane  surfaces  of  any 
solid,  or  for  surfaces  having,  when  related  to  a  given  line  on 
such  surface,  a  curvature  in  one  direction  only.  In  general, 
it  may  be  stated  that  any  solid  may  be  developed  on  whose 
surfaces  it  is  possible  to  lay  a  straightedge,  in  continuous 
contact,  in  any  one  direction.  To  use  in  this  connection 
the  illustration  of  the  cylinder,  it  will  be  seen  that,  if  the 
straightedge  is  resting  on  the  surface  parallel  to  the'axis  of 
the  cylinder,  it  will  remain  in  contact  at  all  points.  If,  on 
the  other  hand,  the  straightedge  is  resting  on  the  curved 
surface  and  is  not  parallel  to  the  axis  of  the  cylinder,  the 
surface  will  be  in  contact  at  a  single  point  only.  However, 
the  fact  that  it  is  possible  to  place  the  straightedge  in  con- 
tinuous contact  on  the  surface  allows  the  inference  that 
such  surface  is  capable  of  accurate  development. 

The  same  rule  applies  to  solids  of  irregular  form.  The 
methods  of  development,  however,  are  not  the  same  in  cer- 
tain variously  formed  solids,  as  will  be  explained  later. 
There  are  certain  forms  Avhose  surfaces,  owing  to  their 
curvature  in  several  directions,  are  not  capable  of  being  thus 
laid  out  on  a  flat  surface,  i.  e.,  not  capable  of  being  devel- 
oped. On  the  surfaces  of  solids  of  this  class— the  sphere,  for 
example — it  will  be  found  impossible  to  lay  the  straightedge 
in  contact  in  any  direction.     For,  if  placed  on  such  a  surface, 


4  DEVELOPMENT  OF  SURFACES  §  1(5 

there  will  be  but  one  point  of  contact — that  of  the  tan- 
gential point.  Tangential  contact  indicates  that  develop- 
ment can  be  accomplished  only  in  an  approximate  way.  For 
purposes  of  development,  then,  it  is  convenient  to  separate 
all  solids  into  two  general  classes  according  to  the  result 
obtained  in  developing  their  surfaces.  These  two  classes 
are:  solids  whose  surfaces  admit  of  accurate  development 
and  solids  whose  surfaces  admit  only  of  approximate  devel- 
opment. Approximate  developments  are,  however,  so 
nearly  accurate  for  the  purposes  of  the  sheet-metal  worker 
that^  the  kind  of  solid  is  more  clearly  marked  by  the  method 
of  developing  its  surface  than  by  the  result  obtained  by  the 
development.  Therefore,  in  order  to  distinguish  the  kinds 
of  solids,  both  accurately  and  approximately  developed 
solids  are  divided  into  three  main  classes  according  to  the 
method  used  in  developing  their  surfaces.  These  classes 
are  explained  later. 

5.  Accurate  Developments. — Solids  whose  surfaces 
are  capable  of  accurate  development  are  of  frequent  occur- 
rence in  the  sheet-metal-working  trades.  To  this  class 
belong  all  prismatic,  cylindrical,  and  conical  forms,  whether 
of  regular  or  irregular  geometrical  form.  It  includes  all 
articles  or  objects  whose  covering  may  be  formed  without 
being  submitted  to  the  operations  known  to  trade  workers  as 
"raising,"  or  '"bumping."  Any  solid  whose  surfaces  may 
be  unrolled  or  spread  out  on  a  flat  surface  without  "  buck- 
ling "  may  be  accurately  developed.  Although  it  is  often 
necessary,  especially  when  working  metal  of  unusual  thick- 
ness, to  take  into  account  the  stretching  of  the  material 
when  producing  patterns  for  many  objects,  these  objects 
belong  to  accurately  developed  solids,  providing  that  the 
metal  does  not  have  to  be  "  raised,"  or  "  bumped,"  in  order 
to  form  the  object.  It  is,  therefore,  essential  that  the 
metal  worker  should  thoroughly  understand  the  nature  of 
the  material  and  be  well  informed  as  to  the  best  manner  in 
which  to  provide  for  all  laps  and  edges  used  in  the  construc- 
tion of  the  finished  article. 


§  IG  DEVELOPMENT  OF  SURFACES  5 

It  is  the  purpose  of  Development  of  Snrfaees  to  define  and 
illustrate  theoretical  developments  and  the  means  used  by 
the  draftsman  in  their  production. 

6,  Approximate  Developiiieuts.  —  The  sphere  and 
other  solids  whose  surfaces  have  a  curvature  in  two  or 
more  directions  are  examples  of  objects  capable  of  only 
approximate  development.  The  test  by  the  straightedge 
is  (with  the  exception  of  the  helicoidal  surface)  a  posi- 
tive indication  of  the  class  to  which  any  solid  may  be 
assigned.  Patterns  for  the  surfaces  of  objects  of  this  class 
may  be  approximated,  because  it  is  necessary  for  the  metal 
to  undergo  the  operations  of  "raising,"  or  "bumping," 
before  it  will  conform  to  the  exact  surface  represented  in 
the  drawing.  It  is  necessary  in  these  cases  to  make  allow- 
ance in  the  pattern  for  the  stretching  of  the  metal.  Since 
this  part  of  the  subject  does  not  belong  to  theoretical  develop- 
ment, it  is  not  treated  here. 


SOIilDS   THAT   MAY    BE    ACCURATELY 
DEYEIiOPED 

7.  There  are  three  distinct  methods  in  common  use,  by 
means  of  which  patterns  are  produced  for  solids  whose  sur- 
faces are  capable  of  accurate  development.  It  is  advisable, 
therefore,  to  separate  the  different  varieties  of  these  forms 
into  three  general  divisions,  in  order  that  their  development 
may  be  studied  in  a  systematic  manner.  This  classifica- 
tion may  be  made  by  studying  the  manner  in  which  the 
covering  of  these  solids — to  use  again  the  illustration  of  a 
wrapper — would  be  unrolled  or  spread  out  if  done  by  roll- 
ing the  solid  on  a  flat  surface. 

8.  Solids  Developed  on  Parallel  Ijines. — A  conve- 
nient illustration  of  the  manner  in  which  the  surfaces  of  a 
solid  will  appear  when  unrolled  as  above  indicated  may  be 
found  in  the  following  example,  which  serves  at  the  same 
time  to  define  a  property  peculiar  to  solids  of  a  certain 
form.     Let  the  continuously  adjacent  surfaces  of  the  prism 


DEVELOPMENT  OF  SURFACES 


16 


shown  in  Fig.  1  (a)  be  carefully  covered  with  thin  paper,  as 
at  Fig.  1  [d).  Denote  each  of  the  four  surfaces  by  a  letter, 
as.^,  B,  C,  and  D,  and  further  designate  the  edges  of  the 
prism  by  the  letters  a  b,  c  d,  e  f,  and  g  h.  As  the  ends  of 
the  paper  covering  meet  at  the  edge  a  b,  that  edge  of  the 
surface  D  may  be  denoted  by  the  letters  a'  b' ,  as  shown  in 


Fig.  1  {b).  Assume  now  that  the  prism  is  laid  on  the 
drawing  board,  the  surface  A  face  down,  and  the  paper 
covering  removed  by  turning  the  prism  over  and  over,  the 
paper  remaining  on  the  surface  of  the  drawing  board,  as 
shown  in  Fig.  2  {a)  and  {b). 

Two  important  principles  relating  to    developments  are 
demonstrated  in  these  illustrations.     First,  as  will  be  seen 


Fig.  2 


from  Fig.  2  {b),  the  edges  a  b,  c  d,  e  f,  g  h,  and  a'  /^' are  all 
parallel  to  one  another.  This  is  true  both  in  the  develop- 
ment and  on  the  solid,  as  may  be  readily  seen  by  the 
student,  the  only  difference  being  that  on  the  solid  certain 
of  the  lines  are  in  different  planes,  while  in  the  development 
they  are  all  in  the  same  plane.      Stro/id,    it  will  be  noted 


§16 


DEVELOPMENT  OF  SURFACES 


that,  since  the  letters  shown  in  Fig  2  are  reversed,  the 
outer  surface  of  the  paper  covering  in  Fig.  1  {U)  corresponds 
to  the  under  surface  in  Fig.  2  {b).  In  a  similar  manner,  it 
is  learned  from  the  second  principle  that  positions  indicated 
on  any  surface  of  a  solid,  as  shown  in  a  projection  drawing, 
are  reversed  when  shown  in  the  development  of  the  solid. 

The  same  treatment  of  the  cylinder  is  found  to  produce 
results  closely  resembling  those  shown  in  the  case  of  the 
prism.      The  cylinder  is  represented  in  Fig.  3  (c?)  as  covered 


Fig.  3 


with  paper,  a  number  of  lines  being  ruled  on  the  covering 
parallel  to  its  axis,  as  shown  at  c  d^  ef,  etc.  The  paper  is 
shown  unrolled  in  Fig.  8  {b),  and  it  will  be  observed  that 
not  only  the  outer  edges  a  b  and  a'  b'  are  parallel  to  each 
other,  but  that  all  other  lines  parallel  to  the  axis  of  the 
solid  appear  in  the  development  parallel  to  one  another  and 
to  the  edge  lines  a  b  and  a'  b' .  The  student  that  has  ever 
flattened  out  a  piece  of  straight  molding,  as  for  cornice 
work,  probably  noticed  a  number  of  straight  parallel  lines 
on  the  metal  where  it  had  been  bent  in  the  brake.  This  is 
an  illustration  similar  to  that  of  the  cylinder,  the  different 
members'  of  the  molding  being  considered  as  the  various 
surfaces  of  an  irregular  solid. 

Such  illustrations  indicate  that  parallel  lines  bear  some 
relation  to  certain  forms,  and  it  will  be  shown  that  the  pat- 
terns for  these  forms  are  developed  by  a  method  whose 
principles  are  based  on  this  fact.  Many  solids  may  be  at 
once  recognized  as  belonging  to  this  division.      In  general^ 


8 


DEVELOPMENT  OF  SURFACES 


§16 


any  solid  whose  edges  are  parallel  may  be  located  here.  In 
the  case  of  solids  having  curved  surfaces,  it  may  be  stated 
that,  if  it  is  possible  to  draw  a  series  of  parallel  lines  on 
such  surfaces,  the  development  of  the  solid  may  be  pro- 
duced by  the  same  methods  given  for  this  class.  The  first 
general  division,  therefore,  comprises  those  solids  whose 
surfaces  may  be  developed  on  parallel  lines. 

9.     Solids    Developed   on    Radial    JLiues.  —  When    the 
test  given  to  the  cube  and  the  cylinder  in  Figs.  1  to  3  is 


Fig.  4 

applied  to  the  pyramid,  it  is  found  that  the  lines  indicated 
on  the  paper  converge  to  a  point,  as  shown  in  Fig.  4  {a) 
and  {b).  It  is  noticed,  also,  that  this  point  o.  Fig.  4,  defines 
the  position  of  the  vertex  of  the  pyramid.  The  same  may  be 
said  of  the  cone,  illustrated  in  Fig.  5   (<^)  and  {b).      If  lines 


Fig.  5 


are  first  indicated  on  the  surface  of  the  cone  corresponding 
to  its  elements,  it  will  be  found,  when  the  covering  is 
unrolled,  that  these  lines  also  converge  to  a  point,  as  in  the 
case  of  the  edges  of  the  pyramid. 


§  16  DEVELOPMENT  OF  SURFACES  9 

It  was  found  possible  to  institute  a  system  of  obtaining 
developments   based    on    parallel    lines    in    the  case  of    the 
prism  and  cylinder;  in  a  similar  manner,  it  is  quite  evident 
in  this  case  that  a  system  dealing  with  radial  lines  should 
produce  like  results.      Since,  in  projection  drawing,  the  ele- 
ments of  the  cone  are  known  to  be  useful  factors  in  deter- 
mining the  position  of  points  on  its  surface,  it  may  readily 
be  conceived  that  their  use  in  a  somewhat  similar  way  may 
be  adapted  to  developments.      This  is  found  to  be  the^  case- 
and  a  second  general  division  of  solids  is  thus  made,  con- 
sisting of  those  forms  whose  surfaces  may  be  developed  on 
radial  hues.      Included  in  this  division  are  all  regular  taper- 
ing solids  and  such    irregular  forms    as  are  derived    from 
regular  solids.      The  metal  trades  furnish  manv  examples  of 
solids  belonging  to  this  division;  in  fact,  the  writers  of  sev- 
eral works  on  patterncutting  confine  their  instruction  almost 
entirely  to  the  development  of  solids  of  this  character. 

10.     Soliils    Developed  by  Triaugnlatlon.— There  are 

many  forms  of  irregular  surfaces  to  which  the  test  of  the 
straightedge  may  be  applied  and  the  conclusion  thereby 
reached  that  their  surfaces  admit  of  accurate  development. 
It  may  also  be  concluded  that  neither  of  the  two  former 
methods  is  applicable,  for  neither  parallel  lines  nor  a  series 
of  radial  lines  may  be  drawn  on  their  surfaces.  Many  of 
these  solids  are  not  of  such  a  shape  as  to  admit  of  their  being 
either  turned  or  rolled  on  a  plane  surface.  It  is  found"^" 
however,  that  on  every  such  surface,  series  of  two  or 
more  lines  each  may  be  drawn  in  certain  directions,  forming 
angles. 

On  such  irregular  surfaces  it  may  happen  that  no  two  of 
the  angles  thus  drawn  on  the  solid,  or  represented— either 
correctly  or  foreshortened— in  the  projection  drawing,  will 
he  in  the  same  plane  or  be  equal  to  each  other.  Since' it  is 
possible  thus  to  project  these  angles,  evidently  thev  may  be 
reproduced  on  the  flat  surface  of  the  drawing  paper  in  their 
correct  size.  If  this  can  be  done,  it  may  be  reasonably 
assumed  that  the  surfaces  thus  represented  will  be  the  same 


M.  E.    /-.- 


-yy 


10       DEVELOPMENT  OF  SURFACES      §  16 

as  the  corresponding  surfaces  of  the  solid.  An  illustration 
of  this  principle,  as  pertaining  to  a  plane  surface,  was  given 
under  another  heading  in  Practical  Projection. 

In  Fig.  6  an  irregular  solid  of  this  kind  is  shown.  It  is 
the  solid  whose  projection  was  drawn  in  Problem  19  of 
Practical  Projection.  This  figure 
illustrates  in  a  general  way  the 
method  used  in  arranging  the  trian- 
gles on  the  irregular  surface  of  such 
solids.  The  triangles  are  represented 
in  the  figure  in  a  perspective  way, 
but  they  are,  of  course,  always  drawn 
in  connection  with  the  usual  methods 
of  projection.  The  third  general  division,  therefore,  con- 
sists of  those  solids  whose  surfaces  are  developed  by  trian- 
gulalion — that  is,  by  means  of  triangles. 

1 1 .  IIoTv  tlie  Division  of  Solids  Is  Acconiplishe<l. 
It  is  not  to  be  understood  that  the  draftsman  actually 
applies  the  test  of  the  straightedge  in  reaching  a  conclusion 
as  to  whether  the  surfaces  of  a  solid  may  be  accurateh*  or 
approximately  developed.  Nor  does  he  roll  the  object  on 
the  drawing  board  in  order  to  determine  whether  the 
method  by  parallel  hnes  or  one  of  the  other  methods  is  to 
be  used.  As  a  matter  of  fact,  he  seldom  has  a  model  to 
work  from,  and,  therefore,  could  not  apply  such  a  test  if  he  so 
desired.  But  as  he  studies  the  drawings  and  imagines  the 
position  of  the  surfaces  as  they  wiU  appear  in  the  com- 
pleted object,  he  is  enabled  to  apply  the  tests  as  ejfectually, 
in  an  imaginary  way,  as  though  the  tests  were  made  with  a 
straightedge.  In  the  same  imaginary  way,  also,  he  assigns 
the  solid  to  the  general  division  to  which  it  properly  belongs, 
and  thus  decides  as  to  the  method  he  will  use  in  the  devel- 
opment of  its  surfaces. 

A  little  practice  will  enable  the  student  to  classify  the 
variously  formed  objects  in  this  way  and  to  select  the 
method  that  shall  be  applied  in  any  given  case.  A 
very  important  part  of    the   pattemcutter's  acquirements 


§16 


DEVELOPMENT  OF  SURFACES 


11 


consists   in    being    able    to    recognize    in   various    irregular 

objects  those  forms  that  may  be 
only  a  portion  of  some  regular 
solid.  In  other  words,  the  student 
must  learn  to  establish  in  his  own 
mind  the  connection  between  com- 
plete and  perfectly  formed  solids 
and  those  objects  in  which  only 
a  portion  of  the  solid  may  be  rep- 
resented. The  method  of  devel- 
opment is,  of  course,  the  same  in 
both  cases,  but  as  a  matter  of  fact. 


Fig. 


Fig.  8 


the  operations  are  usually  more  complicated  in  cases  where 
the  incomplete  solid  demands  the  patterncutter's  attention. 
Especially  is  this  true  of  conical  forms,  or  those  developed 
on  radial  lines.  Frequent  illustrations  of  this  principle 
may  be  found  in  commonly  occurring  objects.  The  flaring 
pail  shown  in  Fig.  7  is  seen  to  be  a  part  (or  frustum)  of  a 
cone,  the  completed  cone  being  indicated  by  the  light  sha- 
ding in  the  illustration. 


12 


DEVELOPMENT  OF  SURFACES 


§  16 


The  same  is  true  of  the  sitz  bathtub  in  Fig.  8.  Here  it 
is  seen  that  the  portion  of  the  cone  represented  by  the  fin- 
ished article  is  an  irregular  section 
of  the  cone ;  its  development  is,  how- 
ever, accomplished  by  the  same  meth- 
ods to  be  shown  for  regular  cones. 
Another  instance  is  found  in  the 
measure  shown  in  Fig.  9.  Here  are 
two  intersecting  cones  :  a  regular 
frustum  of  one  forms  the  body  of 
the  measure;  and  an  irregular  frus- 
tum of  the  other — an  inverted  cone — 
forms  the  lip  of  the  finished  article. 
All  the  articles  in  Figs.  ?  to  9  are 
thus  shown  to  be  frustums  of  regu- 
lar cones,  although  varying  in  the 
regularity  of  their  bases.  In  certain 
cases,   as    in    the    "oval"  pan   body 


Fig.  9  Fig.  10 

represented  by  the  heavily  shaded  band  in  Fig.  10,  the 
surfaces  may  be  portions  of  the  surfaces  of  different  cones  or 
of  cones  differing  in  size.  The  bases  of  this  article  are 
elliptical  in    outline.      The    ellipses    are  drawn  by  circular 


§  IG      DEVELOPMENT  OF  SURFACES       13 

arcs.  The  vertexes  of  the  different  cones  would  be  repre- 
sented in  a  plan  view  by  the  centers  from  which  the  differ- 
ent arcs  are  struck.  These  cones  are  partially  shown  in 
Fig.  10,  and  in  the  relation  required  by  the  portions  of 
their  surfaces  that  compose  the  sides  of  the  pan. 

It  is  essential,  therefore,  that  the  student  should  possess 
a  certain  familiarity  with  the  forms  of  the  regular  solids,  to 
assist  him  in  the  classification  of  the  objects  that  he  will  be 
called  on  to  develop.  It  is  with  this  end  in  view  that  a  series 
of  plates  is  to  be  drawn  by  the  student.  The  instruction 
is  in  the  form  of  problems,  and  several  of  the  drawings 
of  Practical  Projccticm  are  reproduced  for  the  purpose  of 
showing  the  development  of  the  surfaces  of  the  solids  there 
represented.  The  student's  attention  will  first  be  directed 
to  a  consideration  of  those  solids  whose  development  may 
be  accomplished  by  means  of  parallel  lines. 


DEVEILOPMEXT   BY   PARALLEIi   I.I]S:ES 

VI,  Importance  of  Certain  Views. — It  has  already 
been  stated  that  the  making  of  a  working  drawing  is  the 
draftsman's  first  step  towards  obtaining  a  pattern  for  the 
surfaces  of  any  solid.  The  solid  in  question  should  be  shown 
in  this  drawing  in  such  a  position  that  measurements  may 
be  taken  of  its  surfaces  in  all  their  dimensions.  In  order  to 
accomplish  this,  several  views  may  have  to  be  drawn, 
although  a  right  plan  and  elevation  will  usually  be  sufficient. 
In  some  cases  there  may  be  given  to  the  mechanic  a  draw- 
ing in  which  the  object  is  so  shown  that  these  dimensions 
may  not  be  readily  obtained.  In  such  cases,  operations  in 
projection  are  required;  with  these  the  student  is  already 
familiar. 

For  purposes  of  development  it  is  important  that  the  view 
shown  should  be  that  one  in  which  the  lines  of  the  solid  are 
given  in  their  true  length,  or,  in  other  words,  a  right  view 
of  the  solid.  In  addition  to  this  view  there  must  be  given 
that  view  also  in  which  the  surfaces  thus  partially  bounded 


14       DEVELOPMENT  OF  SURFACES      §  16 

by  these  parallel  lines  are  shown  as  07i  edge.  In  some 
instances,  as  in  the  case  of  a  simple  solid  of  which  all  the 
dimensions  are  known,  the  latter  view  may  be  omitted;  it 
is,  however,  understood  as  being  drawn,  for  the  draftsman 
knows  all  its  dimensions.  Generally,  therefore,  before  the 
pattern  can  be  produced,  a  plan  and  an  elevation  showing 
the  solid  in  a  right  position  must  be  drawn. 

In  drawing  the  projections  in  Practical  Projection,  the 
right  views  have  first  been  drawn  and  inclined  views  have 
then  been  projected  from  them.  This  has  been  done  in 
order  to  familiarize  the  student  with  the  appearance  of  such 
drawings;  but  in  every  case  the  development  of  the  pat- 
tern is  to  be  projected  from  a  right  view.  It  is  possible  for 
the  draftsman  to  become  so  expert  by  practice  that  in  cer- 
tain cases  he  is  enabled  to  obtain,  from  foreshortened  views, 
patterns  for  some  surfaces.  The  beginner  should  not  attempt 
this,  however,  since  the  operations  involved  are  confusing, 
and  should  be  resorted  to  only  by  the  experienced  pattern- 
cutter  that  thoroughly  understands  the  subject. 

13.  Tlie  Stretclioiit.  —  As  stated  in  the  preceding 
article,  it  is  essential  that  in  all  cases  where  the  develop- 
ment of  solids  may  be  accomplished  on  parallel  lines,  the 
view  showing  certain  surfaces  as  on  edge  should  either  be 
given  or  assumed.  From  such  a  view,  the  width  of  each 
surface  may  readily  be  ascertained.  The  total  width  of  all 
these  surfaces — the  distance  around  the  solid — is  called  the 
girth-  of  the  solid.  In  case  the  solid  has  a  curved  surface, 
its  girth  is  found  by  spacing  with  the  dividers  the  outline  in 
that  view.  The  girth  of  the  cylinder,  for  example,  is  equal 
to  the  length  of  the  circumference  of  a  circle  that  represents 
the  base  of  the  cylinder. 

When  a  distance  corresponding  to  the  girth  of  any  solid 
is  represented  by  a  straight  line  on  a  fiat  surface,  such  a  line 
is  called  a  stretchout  for  the  development,  or  pattern. 
This  line  is  then  marked  off  by  a  series  of  points,  the  points 
representing  the  places  at  which  the  line  would  be  bent 
if  formed  up  to  correspond  with  the  outline  of  the  solid 


§  IG      DEVELOPMENT  OF  SURFACES       15 

represented  in  the  view  from  wliich  the  distanees  were  taken. 
In  the  case  of  curved  surfaces,  a  number  of  points  are 
located  on  the  outline,  as  previously  indicated.  This  is 
usually  done  by  dividing  the  outline  into  a  number  of  equal 
spaces  in  the  same  manner  as  in  projection  drawing;  an 
equal  number  of  spaces  is  then  stepped  off  on  the  stretchout 
line,  whose  total  length  is  in  all  cases  equal  to  the  girth  of 
the  solid.  An  important  point  to  be  observed  is  that  the 
points  thus  located  on  the  stretchout  must  be  (although 
reversed)  in  a  position  on  the  line  corresponding  to  that 
relatively  occupied  on  the  solid. 

14.  Position  of  the  I)eveloi)nient. — The  position  in 
which  the  stretchout  is  placed  on  the  drawing  determines 
the  position  of  the  development.  This  line  is  always  drawn 
at  right  angles  to  the  parallel  lines  of  the  solid  and  from  a 
view  in  which  these  parallel  lines  are  shown  ///  their  true 
length. 

In  making  a  projection,  then,  from  which  to  produce  the 
patterns  of  any  object,  it  is  important  that  a  sufficient 
space  be  left  on  the  drawing,  to  one  side  or  the  other  of  that 
view.  It  frequently  happens  that  this  cannot  be  done,  and 
in  such  cases  it  is  a  common  practice  to  lay  an  extra  piece 
of  paper  over  a  portion  of  the  drawing,  on  which  the  develop- 
ment may  be  produced.  When  the  latter  method  is  adopted, 
'  the  paper  on  which  the  development  is  made  may  be  used 
in  transferring  the  outline  of  the  pattern  to  the  metal,  and 
the  original  drawing  may  then  be  preserved  in  perfect  con- 
dition. 

15.  Development  of  tlie  Cube. — For  the  purpose  of 
explaining  to  better  advantage  the  use  of  the  stretchout,  the 
development  of  the  cube  is  presented,  step  by  step,  in  Figs.  11 
to  14,  inclusive.  A  reproduction  of  this  development  should 
be  made  by  the  student  in  accordance  with  the  following 
instructions,  although  the  drawing  is  not  to  be  sent  to  the 
Schools  for  correction. 

The  plan  and  the  elevation  shown  in  the  left-hand  portion 
of  the  figures  are  drawn  first,      '^he  development  could  be 


16       DEVELOPMENT  OF  SURFACES      |  16 

produced  from  either  view  in  this  case,  since  in  any  right 
view  the  dimensions  of  a  cube  are  equal.  For  the  purpose 
of  the  illustration,  the  development  is  drawn  from  the 
elevation.  At  right  angles  to  the  parallel  lines  in  the  eleva- 
tion draw  a  line  as  J/ A',  Fig.  11.  On  this  line  locate  a 
point  at  any  convenient  distance,  as  at  iv.  This  point  may 
correspond  to  any  of  the  upright  lines  or  edges  of  the  solid 


Elevation 


it- 


Fig.  11 


represented  by  A,  B,  C,  or  D  in  the  plan  of 
the  cube.  Since  it  is  necessary  to  begin 
from  one  of  these  edges  to  unfold  in  an 
imaginar}-  way  the  surfaces  of  the  cube,  the 
point  zi'  will  be  considered  as  representing 
a  position  on  the  edge  A.  The  surfaces 
are  to  be  represented  in  their  regular  order 
in  the  development,  that  is,  the  order  in  which  they  appear 
on  the  solid  itself;  first,  the  surface  represented  in  the  plan 
by  A  B,  then  B C,  CD.  and  DA.  in  their  natural  order  as 
they  are  shown  in  the  projection  at  the  left  of  the  line  MX. 
The  dividers  may,  therefore,  be  set  at  a  distance  equal  to 
the  length  of  the  side  A  B.  and  since  the  sides  of  the  cube 
are  equal  in  length,  the  distances  zl'x,  xy,Y3,  sltvAzw' — 
corresponding,  respectively,  to  the  sides  represented  in  the 
plan  by  A  B,  B  C.  CD,  and  DA — are  to  be  spaced  off  on 
the  line  MX. 

16.  Laying  Off  tlie  Stretchout.  —  That  portion  of 
the  line  J/ A' included  between  the  points  iv  and  xv  is  called 
the  stretchout  of  the  cube.  A  stretchout  may  be  drawn  in 
any  position  on  the  drawing  board,  at  the  convenience  of  the 
draftsman,  but  it  is  invariably  at  right  angles  to  the  parallel 
lines  of  the  solid. 

Wherever  the  stretchout  occurs  in  the  drawings  of  this 
section,  it  is  represented  by  a  heavy  line,  as  shown  in  Fig.  11. 
It  is  customary  to  draw  a  fine  of  indefinite  length  quite  near 


16 


DEVELOPMENT  OF  SURFACES 


the  view  of  the  solid  that  is  being  developed,  as  in  Fig.  n. 
When  the  stretchout  is  mentioned,  the  only  part  of  the  line 
referred  to  is  that  included  between  the  extreme  points  zv 
and  w',  Fig.  H,  located  to  define  the  total  width  of  the 
adjacent  surfaces.  This  operation  is  called  laying  off,  or 
developing,  the  stretchout.  It  will  be  seen  that  if\  string 
equal  in  length  to  %v  w'  should  be  stretched  around  the  cube 
in  a  horizontal  direction,  it  would  exactly  reach  the  entire 
distance,  and  the  ends  Avould  meet,  in  Fig.  11.  at  the 
edge  A. 

The  next  step  is  to  erect  perpendiculars  to  the  stretch- 
out MN  that  shall  pass  through  the  points  u',  x,  j,  etc. 
and  be  produced  on  both  sides  of  the  line.  This  is  done  by 
means  of  the  triangle,  in  connection  with  the  T  square,  as 
in    Fig.    12.      The    lines  thus   drawn  are  called  edge  lines, 


Fig.  12 

since  they  represent,  in  the  development,  those  portions  of 
the  surfaces  that  would  form  the  edges  of  the  solid  if  the 
pattern  were  to  be  cut  out  and  formed  up  to  the  shape 
indicated  by  the  projections.  Edge  lines  are  to  be  repre- 
sented in  these  drawings  by  dash-and-dot  lines,  as  shown  in 
Fig.  12,  similar  to  those  used  in  Praetical  Projection  for 
projectors. 


18       DEVELOPMENT  OF  SURFACES      §  16 

Next,  the  length  of  each  of  the  upright  edges  shown  in 
the  elevation  is  marked  off  on  its  corresponding  edge  line 
in  the  development.  This  is  readily  done  with  the  T  square, 
and,  since  each  of  the  four  edges  represented  in  the  elevation 
is  of  the  same  length,  the  T  square  may  be  brought  even 
with  the  horizontal  lines  in  that  view,  and  a  line  drawn 
from  each  across  the  development,  as  shown  in  Fig.  13. 
The  lines  thus  drawn  are  called  developers,  and  will  be  repre- 
sented in  these  drawings  by  broken  lines,  as  shown  in  Fig.  13. 
This  name  is  applied  to  them  for  the  reason  that  the  length 
of  the  edge  lines  is  determined — developed — when  a  developer 
is  drawn  from  each  extremity  of  the  edge  shown  in  the'eleva- 
tion  to  its  corresponding  edge  line  in  the  development.  In 
this  case,  the  four  upright  edges  of  the  cube,  represented 
by  the  two  vertical  lines  in  the  elevation,  are  of  the  same 
length,  and  their  ends  are  in  the  same  horizontal  lines; 
therefore,  the  two  developers  /  </  and  r  s,  drawn  from  the 
«'  b'  c'  d  a 

£± I .^ J J L. 


El-e^vation 


elevation,  define  the  upper  and  lower  bound- 
aries of  each  of  the  four  upright  surfaces  of 
the  cube. 

The  development  of  four  sides  of  the  cube 
is  thus  accomplished;  and  if  desirable,  the 
other  two  sides  may  be  added  to  any  one  of  the  four  that 
have  been  developed.  Since  the  method  of  obtaining  a 
development,  as  it  is  now  called,  or  a  full  view  of  any  side 
or  surface  of  a  solid  is  already  familiar  to  the  student,  no 
further  explanation  will  be  given.  In  sheet-metal  work  it 
is  generally  preferable  to  make  the  ends  (of  forms  similar 
to  the  cube)  of  separate  pieces  of  metal,  and.  on  account 
of  the  waste  of  stock,  it  Avill  seldom  be  found  desirable  to 
combine  all  the  surfaces  of  a  solid  in  a  single  piece.      Should 


16 


DEVELOPMENT  OF  SURFACEvS 


19 


such  a  case  arise,  however,  the  full  view  would  be  pro- 
jected and  afterwards  copied  into  its  proper  place  on  the 
development. 

The  development  of  any  solid  of  this  class,  whose  bases 
are  parallel  and  at  right  angles  to  its  parallel  lines,  is  always 
a  parallelogram ;  and,  as  in  the  development  of  the  cube  in 
Fig.  13,  this  is  divided  into  smaller  parallelograms,  each 
representing  a  surface  of  the  solid. 

17.  Finishing  the  Drawing. — In  order  to  enable  the 
draftsman  to  distinguish  the  features  of  a  development  at  a 

'i. b'  e'  ft'  /.' 


Elevation 


Plan 


ST. 


T 


T 


j£. 


\ 


4 


-K 


Fig.  \i 


glance,  it  is  customary  to  define  the  outer 
edges  of  such  a  drawing  by  means  of  full 
lines,  as  is  shown  in  Fig.  14.  The  outer 
edges  are  further  distinguished  by  small 
arrowheads,  while  the  other  edge  lines  of 
the  pattern  are  marked  near  each  extrem- 
ity by  a  small  circle  drawn  by  freehand  methods,  in  the 
manner  shown,  thus  indicating  to  the  mechanic  that  the 
sheet  is  to  be  bent  along  this  line.  It  is  sometimes  desir- 
able, as  in  detail  developments  for  certain  classes  of  sheet- 
metal  work,  to  designate  the  stretchout  by  a  line  drawn 
with  a  blue  pencil,  thus  readily  attracting  the  draftsman's 
attention. 

The  mechanic  seldom  resorts  to  the  drawing  board  in 
order  to  produce  a  development  of  a  simple  solid  such  as 
the  cube,  since  the  same  result  may  be  accomplished  with 
the  steel  square,  the  sizes  being  marked  out  directly  on  the 
metal.  The  development  of  the  cube,  however,  has  been 
shown  in  these  illustrations,  inasmuch  as  by  the  same  princi- 
ples any  solid  of  this  class  may  be  developed.  It  may  also 
be  stated  that  the  draftsman  rarely  represents  developers 


20 


DEVELOPMENT  OF  SURFACES 


§  16 


or  edge  lines  by  the  particular  lines  used  for  that  purpose 
in  this  section.  These  distinguishing  lines  have  been 
adopted  here  solely  for  the  purpose  of  fixing  clearly  in  the 
student's  mind  the  principles  on  which  these  drawings  are 
made.  After  these  principles  have  been  mastered,  the  use 
of  such  lines  in  practical  work  may  be  discontinued,  and 
the  student  may  then,  by  the  use  of  light  pencil  lines  only, 
proceed  with  the  development  of  such  other  solids  as  he 
ma}-  be  called  on  from  time  to  time  to  lay  out.  These 
drawings,  when  inked  in,  should  be  completed  in  the  man- 
ner shown  in  the  illustrations. 

18.     Development    of   Intersected    Solids. — In    cases 
where  the  parallel  lines  of  a  solid  are  interrupted   by  the 


b  c  d 

intersection  of  another  solid  or  by  a  cut- 
ting plane,  it  becomes  important  to  follow 
carefully  the  instructions  just  given;  in 
such  cases  it  is  of  extreme  importance 
that  an  exact  development  should  be 
made  on  the  drawing  board.  For  exam- 
ples of.  the  development  of  such  inter- 
sected solids,  we  will  refer  to  the  figures  illustrating  the 
cutting  planes  in  their  effect  on  the  cube,  which  figures 
have  become  familiar  to  the  student  from  their  use  in  Prac- 
tical Projection.  Fig.  15  is  a  reproduction  of  one  of  these 
projections,  showing  the  development  of  the  parallel  sur- 
faces. Here  it  will  be  seen  that  the  development  can  be 
produced  only  from  the  elevation,  since  the  cutting  plane 
has  altered  the  solid  in  such  a  manner  as  to  admit  of  par- 
allel lines  being  drawn  in  but  one  direction. 

The  stretchout  is  drawn  as  before  and  the  width  of  the 
surfaces  spaced  off  in  the  usual  manner.  It  will  be  further 
noticed  that,-  since  the  edges  of  the   cube   are  unequal  in 


§  16      DEVELOPMENT  OF  SURFACES       21 

length,  it  becomes  more  important  to  observe  the  order  of 
the  surfaces  as  they  are  being  unrolled  from  the  solid. 
After  the  edge  lines  are  drawn  in  the  development,  the 
developers  are  drawn  in  the  same  manner  as  in  Fig.  13,  but 
with  this  difference:  the  lower  ends  of  the  edge  lines  are 
defined  by  a  single  developer  as  before,  but  it  becomes 
necessary  to  draw  a  developer  from  the  upper  end  of  each 
edge  in  the  elevation  to  its  corresponding  edge  line  in  the 
development.  If  this  is  done  carefully,  it  will  be  seen, 
from  a  comparison  of  the  surfaces  in  the  development  with 
those  on  the  solid  in  the  elevation,  that  they  are  in  the 
same  relative  position  with  reference  to  one  another, 
although  reversed.  This  is  clearly  indicated  in  Fig.  15  by 
the  use  of  similar  capitals  and  small  letters  for  the  corre- 
sponding edges  and  edge  lines,  respectively,  in  the  projec- 
tion and  development.  Thus,  the  edge  line  a  a  represents 
the  edge  A\  b  b'  represents  the  edge  i),  etc.  It  will  be 
noticed,  also,  that  those  parts  of  a  development  that  come 
together  and  form  edges  or  seams  are  indicated  by  the  same 
letters.  A  similar  principle  of  lettering  these  drawings 
will  be  continued  throughout  Development  of  SiD'faees, 
since,  when  once  understood  by  the  student,  he  can  study 
the  drawings  intelligently  and  with  less  reference  to  the 
descriptive  text. 

In  this  drawing  of  the  cube,  another  fact  is  presented 
that  demands  care  on  the  part  of  the  draftsman ;  that  is, 
the  outer  edge  lines  in  the  development  must  be  of  the  same 
length.  It  may  seem  unnecessary  to  call  attention  to  a  fact 
so  obvious,  since  it  is  very  clear  that,  as  the  outer  edge 
lines  represent  the  same  edge  of  the  solid,  they  must,  there- 
fore, be  of  the  same  length  in  the  development.  It  is, 
however,  a  cause  of  frequent  error,  and  is  due  simply  to 
carelessness  in  drawing  the  developer  to  the  wrong  edge 
line.  Great  care  must  be  exercised  in  this  respect,  since  the 
accuracy  of  a  development  depends  in  no  small  degree  on 
this  feature. 

A  development  similar  to  the  one  given  in  Fig.  la  is 
shown  in  Fig.  l<j.     Tiiis  development  is  made  from  the  front 


22  DEVELOPMENT  OF  SURFACES  §  16 

elevation  of  the  figure  in  Practical  Projection  that  shows  how 


a  plane  that  cuts  a  cube  is  represented. 
Fig.  15  shows  a  development  made  from 
the  side  elevation  of  the  same  figure.  An 
excellent  illustration  is  here  furnished  of 
the  importance  of  all  edges  of  a  solid  be- 
ing defined  in  that  view  from  which  a 
development  is  made.  The  drawing  in 
Fig.  15  represents  the  cube  in  such  a  position  that  the 
lengths  of  all  its- parallel  edges  are  shown,  while  in  Fig.  16. 
the  length  of  the  edge  B  is  seen  only  by  the  aid  of  the 
dotted  line.  In  drawings  of  this  class,  therefore,  all  edges 
should  be  indicated,  whether  on  the  side  nearest  the  observer 
or  not.  In  such  drawings,  however,  it  is  customary  to  rep- 
resent these  hidden  edges  by  dotted  lines,  in  order  to  avoid 
confusion. 

It  frequently  happens  that  the  cutting  plane  so  intersects 
the  surfaces  of  a  solid  as  to  produce  angles  at  points  other 
than  at  the  vertical  edges  of  the  solid.  An  example  of 
this  is  found  in  Fig.  IT.  The  method  of  obtaining  the 
development  is,  in  the  main,  similar  to  that  used  in  the  pre- 
ceding cases.  From  the  plan  of  the  cube  in  Fig.  IT.  how- 
ever, it  will  be  seen  that  points  are  indicated  on  the  lines 
B  C  and  C  D  denoting  the  corners,  or  points,  at  the  extrem- 
ities of  the  line  K  L.  The  distances  B  K  and  D  L  must, 
therefore,  be  indicated  on  the  stretchout  MX,  as  shown  at 
X  k  and  /  z,  the  points  in  each  case  being  located  from  b  b' 
and  dd'  towards  c c' .  This  is  because  the  points  K  and  L 
approach  C  in  the  plan  in  their  distances  from  B  and  D, 
respectively;  according  to  the  foregoing  instruction,  it  is 
necessary  to  define  them  in  a  position  in  the  development 
corresponding  to  that  represented  on  the  solid. 


§16 


DEVELOPMENT  OF  SURFACES 


23 


Parallel  lines  must  be  produced  throuoh  the  points /{•  and  / 
in  the  same  manner  as  the  edge  lines  were  drawn.  In  a 
certain  way,  these  lines  serve  a  similar  purpose,  since  they 
are  the  termination  of  certain  developers;  but  as  it  is  not 


Fig.  17 


necessary  to  bend  the  surfaces  of  the 
pattern  on  such  lines,  they  are  distin- 
guished from  the  regular  edge  lines  by 
the  term  intcrcdtrc  Hues.  Interedge  lines 
are  as  essential  in  determining  the  out- 
line of  a  development  as  the  edge  lines 
themselves,  and  are  to  be  distinguished  on  the  drawings  for 
this  section  by  the  dash-and-double-dot  line.  A  comparison 
of  the  projections  with  the  development  in  Fig.  17,  made  by 
looking  for  similar  capitals  and  small  letters  in  the  figure, 
will  enable  the  student  to  see  clearly  the  manner  in  which 
the  additional  spaces  are  located  on  the  stretchout,  and  also 
how  this  development  is  produced  from  the  elevation. 

19.  Importance  of  Accuracy. — The  attention  of  the 
student  has  frequently  been  called  to  the  necessity  for  accu- 
racy in  his  drawings.  If  this  is  necessary  in  the  case  of 
projection  drawing,  it  is  doubly  important  in  the  case  of 
developments.  Too  much  stress  cannot  be  laid  on  this  very 
important  feature  of  the  patterncutter's  training.  Unless 
his  drawings  are  accurate  they  are  of  no  value;  it  is.  there- 
fore, of  the  utmost  importance  that  the  patterns  for  any 
piece  of  sheet-metal  work  should  be  carefully  and  accurately 
described.  The  draftsman  may  thoroughly  understand  all 
the  principles  involved  in  projection  drawing  and  the  devel- 
opment of  surfaces;  but  if  the  work  on  the  drawing  board 
has  been  done  in  a  careless  manner,  the  pattern  is  as  liable 


24       DEVELOPMENT  OF  SURFACES      §  16 

to  be  incorrect  as  though  it  had  been  '' guessed  at,"  or  "cut 
and  trimmed." 

There  are  few  solids  whose  development  may  be  accom- 
plished by  the  aid  of  the  limited  number  of  lines  required  in 
the  case  of  the  cube.  The  same  principles,  however,  govern 
all  solids  of  this  class,  and  it  is  necessary  merely  to  be  careful 
and  observe  the  form  of  the  solid  as  it  is  shown  in  the  pro- 
jection drawing.  The  fact  that  the  drawing  contains  many 
lines  should  not  deter  the  student  from  recognizing  each 
surface  independently  of  the  others,  although  it  may  require 
more  care  on  his  part  to  select  the  correct  lines  in  each  case. 

30.  Tlie  Imafirinatiou  a  Great  Help. — The  student" s 
imagination  will  be  found  to  be  his  best  assistant  in  this 
work,  and  by  the  aid  of  the  projections  he  should  picture  to 
himself  a  model  of  the  object  represented.  Further,  he  will 
find  it  a  valuable  help  in  being  able  to  imagine  the  surfaces 
as  they  would  appear  if  a  covering  of  the  solid  were  unrolled 
and  spread  out  on  the  drawing  board  for  the  development. 
In  this  way,  the  corresponding  surfaces  on  the  solid  and  in 
the  development  may  be  compared,  and  the  student  may  be 
able  to  detect  any  errors  that  might  otherwise  escape  him. 


GEXERAL  RUBLES  FOR  OBTAIXI^TG  PAKALUEL. 
DEVELOP3IEXTS 

21.  For  the  convenience  of  the  student,  and  to  aid  him 
in  the  production  of  developments  of  solids  by  means  of 
parallel  lines,  a  general  summary  of  the  important  features 
is  here  presented.  This  summary  contains  the  substance  of 
the  foregoing  instruction. 

1.  A  projection  must  first  be  drawn,  consisting  of  a  plan 
and  elevation,  showing  the  solid  in  a  right  position. 

2.  The  development  is  always  obtained  from  that  view  in 
which  the  parallel  lines  are  shown  in  their  true  lengths. 

3.  The  stretchout  is  drawn  at  right  angles  to  the  parallel 
lines  of  the  solid. 


a  c 


i  i 


M 


Ffff.  /. 


i    i    j 


OEVELD 


cr" 


C 


\s'  //l 


i/-- 


9' 


T~ 


3- 


4 


AT" 


V       e 


/e^- 


-d>' 


a"       i>'  I 


1^    h  i7 


{ 


r. 


JUME  25.  /as  3. 


C;z;.-r-^-:,  ZS-^.  zy  The  C 
AUrirf 


MENT5-I. 


\(c) 


A                                3                      a' 

t 

/V  — 

!    1    ! 

a    3   4   s 

6 

1       1 
7     3"\9 

['""! — ! — 

1 

1            >H 

/4    /_5 

' 

1 

1 

> 

A 

I          4 

3'                  c 
B 

1 

^/>.  /?. 

rV 

a 

ERY  Engineer  Company. 
eserved. 


JOHN  SMITH,  CLASS  A/°  4-529. 


§  IG      DEVELOPMENT  OF  SURFACES       25 

4.  To  indicate  the  width  and  relative  position  of  the  sur- 
faces, points  are  located  on  the  stretchout  corresponding  to 
the  place  of  those  points  in  a  view  that  represents  the  sur- 
faces on  edge. 

5.  Edge  lines  and  interedge  lines  are  always  drawn  at 
right  angles  to  the  stretchout. 

6.  Developers  are  drawn  from  each  edge  or  interedge 
represented  in  the  projection  drawing  to  the  corresponding 
edge  line  or  interedge  line  in  the  development.  The  position 
of  points  located  on  these  lines  is  determined  in  a  similar 
manner. 

7.  Interedge  lines,  when  necessary  for  the  development, 
must  be  indicated  on  the  projection  as  well  as  on  the  develop- 
ment, and  the  same  care  exercised  with  the  corresponding 
developers  as  with  those  drawn  from  edges  to  edge  lines. 

8.  The  length  of  the  outer  edge  lines  in  a  complete 
development  must  be  defined  by  the  same  developers. 

These  instructions  should  be  carefully  observed  by  the 
student,  and  if  the  work  involved  in  the  accompanying  prob- 
lems is  done  in  accordance  with*the  principles  just  enumer- 
ated, no  difficulties  will  be  encountered  that  may  not  be 
readily  overcome  by  careful  study  and  a  comparison  of 
the  drawings  with  these  rules. 


DRAWIXG    PLATE,    TITI.E :    DEVEI.OP3IENTS    I 

22.  For  Development  of  Surfaces  the  student  is  required 
to  draw  five  plates,  which  are  the  same  size  as  those  drawn 
ior  Practical Projcctio)i  and  in  accordance  with  the  same  gen- 
eral instructions.  The  titles  of  the  plates  are  given  and 
are  to  be  placed  and  lettered  in  the  same  manner  as  hereto- 
fore. The  division  lines  between  the  problems  are  to  be 
omitted,  and  in  their  place  a  general  arrangement  is  to 
be  followed  Avhich  resembles  the  reduced  copies  of  the  draw- 
ings shown  on  the  printed  plates.  The  problems  should  first 
M.  v..    v.— 30 


26 


DEVELOPMENT  OF  SURFACES 


S  16 


be  drawn  on  separate  paper  to  the  given  sizes.  The  develop- 
ments may  then  be  drawn  in  such  positions  on  the  plates 
as  to  present  an  appearance  similar  to  that  of  the  reduced 
copies. 

These  problems  consist  mainly  of  developments  of  solids 
whose  projection  occupied  the  attention  of  the  student  in 
the  study  of  Practical  Projection.  They  have  been  selected 
for  this  purpose  because  they  are  representative  solids  whose 
development  affords  an  illustration  of  the  principles  involved 
in  patterncutting.  The  student  that  desires  to  pursue  the 
study  at  greater  length  may  find  convenient  illustrations  in 
other  objects  of  frequent  occurrence.  Desirable  practice 
may  thus  be  obtained,  and  the  practical  application  of  the 
principles  outlined  will  serve  to  fix  them  more  definitely  in 
the  student's  mind.  No  insurmountable  difficulties  are 
likely  to  be  encountered  in  developments  thus  undertaken. 
Additional  work  of  this  kind  should  be  of  the  same  class  as 
the  developments  explained  in  the  text. 


PROBLEM  1 

23.     To  develop  the  stii-faees  of  a  pentagonal  prism. 

A  perspective  view  of  the  prism  is  presented  in  Fig.  18. 
The  drawing  is  to  be  made  on  the  plate 
to  the  size  indicated  by  the  dimension 
figures  there  given.  The  completed  de- 
velopment is  shown  at  Fig.  1  {a)  and  {b) 
on  the  plate.  First  draw  the  plan  in 
the  position  shown  and  then  the  eleva- 
tion, according  to  the  given  sizes.  Xext. 
draw  the  stretchout  M  A',  spacing  oflF  on 
its  length,  as  previously  instructed,  the 
width  of  the  surfaces;  after  this  the 
edge  lines  are  drawn,  and  finally  the  de- 
velopers.    As  before,    indicate   the    edge 

lines,  and  finish  this  and  all  drawings  in  the  manner  described 

in  Arts.  16  to  18, 


Fig.  18 


§  16      DEVELOPMENT  OF  SURFACES       27 

probt^e:m  3 
24.     To  develop  tlie  curved  surface  of  a  cylinder. 

Explanation. — No  perspective  figure  is  given  for  this 
problem,  the  cylinder  being  1^  inches  in  diameter  and 
1^  inches  high.  Since  there  are  no  parallel  edges  presented 
on  the  curved  surface  of  the  cylinder,  it  is  necessary  to 
assume  them.  When  edges  are  thus  assumed  for  the  curved 
surface  of  any  solid,  the  corresponding  edge  lines  in  the 
development  are  represented  in  the  same  manner  as  for  the 
prism.  It  is  unnecessary,  however,  to  designate  assumed 
edge  lines  by  small  circles,  since  there  is  no  angular  bend- 
ing on  such  lines  when  the  surface  is  formed  to  the  shape 
indicated  in  the  projections. 

When  it  becomes  necessary,  on  account  of  the  intersec- 
tion of  another  solid  or  plane  with  the  cylinder,  to  represent 
intermediate  lines,  they  are  then  treated  as  interedge  lines, 
the  same  as  for  the  prism.  Edge  lines  and  interedge  lines, 
therefore,  bear  the  same  relation  to  the  development  of 
curved  surfaces  as  to  plane  surfaces.  But  it  is  necessary  to 
exercise  more  care  in  the  development  of  curved  surfaces, 
since  these  lines  are  not  so  readily  distinguished  from  one 
another  as  in  the  case  of  prisms.  Indicators  may  be  marked 
on  the  outer  edge  lines;  but,  since  the  others  are  assumed 
merely  for  the  purposes  of  development,  the  small  circular 
indicators  are  omitted.  In  the  case  of  a  regular  cylinder,  as 
in  this  problem,  it  is  unnecessary  to  project  the  edge  lines 
to  the  elevation,  and  this  work  may  be  omitted  in  the  con- 
struction of  the  problem  on  the  plate. 

Construction. — -Draw  first  the  plan  and  elevation  shown 
on  the  plate  at  Fig.  2  («),  giving  the  figure  the  required 
dimensions.  Next,  divide  the  outline  in  the  plan  into  a 
convenient  number  of  equal  spaces  (in  this  case  IG).  Draw 
the  stretchout  Af  Nas  heretofore,  and  lay  off  on  this  line  an 
equal  number  of  spaces  similar  to  those  on  the  plan  of  the 
cylinder;  draw  the  edge  lines  as  shown,  and  complete  the 
development  by  drawing  the  two  necessary  developers  from 
the  elevation.  Then  finish  the  drawing  in  the  usual  mnaner, 
as  shown  at  Fig.  2  {d). 


28 


DEVELOPMENT  OF  SURFACES 


16 


PROBLEM  3 

25.     To     develop     the    surfaces     of    an     intersected 
octagonal  j)risni. 

Fig.  19  is  a  perspective  view  of  the  prism  projected  in 
Practical  Projection  in  order  to  show  an  octagonal  prism 
cut  by  a  plane;  the  dimensions  of  the  fig- 
ure, however,  are  slightly  changed,  as  may 
be  seen  from  the  plate.  The  drawing  of 
the  projections  is  similar  to  that  in  the 
problem  referred  to  and  may  be  com- 
pleted as  shown  on  the  plate  at  Fig.  3  {a). 
This  problem  requires  interedge  lines,  as 
in  the  case  of  the  cube  in  Fig.  17;  more 
developers,  too,  are  needed  than  are  used 
for  preceding  problems.  Aside  from 
these  features,  the  drawing  does  not 
differ  materially  from  those  that  have  pre- 
ceded it.  After  developing  the  stretch- 
out, as  in  previous  problems,  and  drawing 
the  edge  lines  and  interedge  lines  at  right  angles  to  the 
stretchout,  developers  are  drawn  from  the  several  points 
indicated  in  the  elevation;  viz.,  E'\  D" ,  C" ,  etc.,  as  shown. 
Care  must  be  used  to  terminate  each  developer  on  the  line 
corresponding  to  each  edge  or  interedge,  as  the  case  may  be. 
After  this  development  is  completed,  the  student  may 
derive  some  assistance  by  cutting  out  a  paper  model  accord- 
ing to  the  lines  indicated  and  bending  it  to  conform  to  the 
octagonal  prism ;  he  should  then  understand  exactly  what  is 
implied  by  the  operations  that  have  been  explained. 


Fig.  19 


PROBLEM  4 

tlie    curved    sui-face 


of   an    inter- 


26.     To    develop 
sected  cylinder. 

Fig.  20  is  a  perspective  drawing  of  the  intersected  cylinder 
required  for  this  development;  the  dimensions  in  Fig.  4  on 
the  plate  indicate  the  size  the  draAving  must  appear  thereon. 


§1G 


DEVELOPMENT  OF  SURFACES 


29 


This  problem  is  very  similar  to  Problem  2,  the  only  difference 
being  that  it  is  necessary  to  project  the  edge  lines  to  the 
elevation  in  order  to  obtain  the  points  of 
their  intersection  with  the  cutting  plane. 
From  these  points  developers  are  drawn 
to  their  corresponding  edge  lines  in  the 
development.  The  curve  traced  through 
points  of  intersection  in  that  portion  of  the 
drawing  is  the  upper  outline  of  the  devel- 
opment. This  drawing  being  fully  shown 
on  the  plate  at  Fig.  4  (cr)  and  (/;),  the  stu- 
dent should  have  no  difficulty  in  comple- 
ting the  problem.  Since  the  cutting  plane 
in  this  problem  is  at  an  angle  of  45°,  the  de-  ^"^  ^" 

velopment  may  be  used  as  a  pattern  for  a  two-pieced  elbow. 


PROBLEM  5 

27.  To  develop  the  surfaces  of  tAvo  intersecting 
.cylindei's. 

Explanation. — Fig.  21  is  a  perspective  view  of  intersect- 
ing cylinders.  Since  the  two  cylinders  are  of  the  same 
diameter,  their  axes  intersecti'ng,  the  lines  of  intersection 
are  represented  on  the  drawing  by  straight  lines. 

Construction. — The  projections  shown  on  the  plate  at 
Fig.  0  (a)  are  first  completed  and  an  end  view  of  the 
shorter  cylinder  projected  as  shown  at  (d). 
The  outline  of  each  cylinder  in  that  view 
in  which  it  is  represented  as  on  edge  is 
then  divided  into  a  similar  number  of 
equal  spaces  (IG  in  Fig.  5).  The  points 
thus  located  for  the  purpose  of  represent- 
ing the  assumed  edges  are  then  projected 
from  each  view  to  the  elevation.  Stretch- 
outs J/.^' and  J/' .\''  are  then  developed, 
t'iG.  21  and  edge  lines  are  drawn  perpendicular  to 

the  stretchout  in  each  development.     Developers  may  now  be 
drawn  from  the  ends  of  all  assumed  edges  in  the  elevation 


30       DEVELOPMENT  OF  SURFACES      §  16 

to  their  corresponding  edge  lines  in  the  developments. 
Note  that  in  the  development  of  the  vertical  cylinder  the 
outline  of  the  intersection  of  the  horizontal  cylinder  is 
projected  to  any  set  of  edge  lines  desired,  the  position 
of  the  outline  of  the  opening  being  optional  with  the 
draftsman. 

The  development  of  the  intersecting  cylinder  could  be 
drawn  from  either  the  plan  or  the  elevation  in  this  case, 
since  the  true  length  of  the  parallel  lines  of  that  solid  is 
shown  in  either  view ;  for  the  sake  of  the  appearance  of  the 
drawing,  it  is  developed  from  the  elevation.  The  develop- 
ment of  the  intersected  solid  is  here  shown  to  be  a  parallelo- 
gram having  an  irregularly  curved  portion  outlined  in  the 
central  part  of  the  figure.  It  may  be  seen  that  the  plan  of 
the  cylinders  is  not  absolutely  necessary  for  this  develop- 
ment, since  the  edge  lines  from  each  circle  intersect  in  the 
same  points  on  the  line  of  intersection  of  the  two  cylinders. 
In  the  practical  work  of  such  developments,  therefore,  one 
of  the  full  views  may  hereafter  be  omitted. 

28.  Reeapitulatiou. — These  five  problems  are  to  be 
copied  on  the  plate  in  the  relative  positions  indicated  on  the 
printed  copj*.  Care  should  be  exercised  that  the  figures 
when  completed  occupy  central  positions  on  the  plate,  and 
an  equal  distance  should  be  left  on  all  sides  of  the  drawing. 
The  plate  will  then  present  a  neat  appearance,  and  the 
developments  ma}^  be  easily  distinguished  from  one  another. 
When  finishing  these  drawings  on  the  plate,  the  various 
features  are  to  be  represented  as  follows: 

Represent  the  boundaries  of  figures  and  all  visible  parts 
of  solids  by  light  full  lines.         

Hidden  edges  and  hidden  intersection  lines  should  be 
represented  by  light  dotted  lines.      

Projectors,  as  heretofore,  are  to  be  indicated  by  dot-and- 
dash  lines.  ■ 

Edge  lines  are  indicated  by  dot-and-dash  lines,  used  also 
for  projectors. 


JUNE  25.  /393. 


Copyright,  1899,  by  THE  Cc 
All  right 


ERY  Engineer  Companv. 
served. 


JOHN  SM/TH,  CLASS  N94529. 


16 


DEVELOPMENT  OF  SURFACES 


31 


Interedge    lines    are    indicated    by    dash-and-double-dot 
lines.  

Represent  developers  by  broken  lines. 

Represent  the  stretchouts  by  heavy  full  lines.    


DRAWi:NrG    J'liATE,   TITLE:     DEVET.OPME]S^TS    II 


PUOllLEM   G 

2^.     To    develop    the    surfaces    of    two    intei'secting 
prisms. 

The  prisms  developed  in  this  problem  are  the  octagonal 
and  quadrangular  solids  whose  projec- 
tion was  given  in  Practical  Projection 
in  order  to  show  views  of  intersecting 
prisms. 

CoNSTRucTiox. — The  development 
of  Case  I  of  that  problem  is  shown  in 
Fig.  22.  Note  that  interedge  lines  are 
required  for  the  development  of  both 
solids,  their  positions  being  found  by 
projectors    drawn    from    the    different 


\     I      I 


i«~vr 


views  of  the  drawing  to  the  full 
view  of  the  end,  as  shown  in 
Fig.  22. 

Case  II  of  the  problem  referred 
to  is  developed  in  a  similar  man- 

FlG.  22  A     1  •  ,-     , 

ner.     A  drawmg  of  the  latter  will 
form  the  first  development  for  this  plate,  and  the  projections 


32       DEVELOPMENT  OF  SURFACES      §  10 

are  accordingly  reproduced  as  shown  at  Fig.  1  (a)  of  this 
plate.  In  order  to  avoid  confusion  of  lines,  projectors  are 
sometimes  drawn  as  shown  on  this  plate ;  that  is,  only  their 
starting  points  are  indicated,  as  between  the  two  views  of 
the  octagonal  prism  in  this  drawing.  Next,  develop  the 
stretchouts  for  the  two  solids,  as  at  J/iV  and  M'  N\  and 
draw  edge  and  interedge  lines  through  their  respective 
points.  The  positions  of  the  interedge  lines  are  found  by 
projecting  the  point  A  in  the  plan  across  to  the  full  view, 
as  shown  at  [d),  afterwards  locating  the  points  x  and  y 
at  x'  and  J''  in  {d');  thence,  they  are  projected  to  the  eleva- 
tion and  carried  to  the  development  in  the  usual  manner, 
the  resulting  figures  at  {b)  and  {c)  completing  the  develop- 
ment of  the  solids. 

The  development  of  Case  III  of  the  problem  in  Practical 
Projection  forms  the  second  part  of  this  problem.  These 
projections  are  reproduced  in  Fig.  2  on  the  plate  at  {a). 
In  this  case,  the  axes  of  the  solids  do  not  coincide,  and  the 
problem  has  an  appearance  of  greater  complication  than 
the  drawing  in  Fig.  1.  It  is  necessary  to  trace  out  the  line 
of  intersection  very  carefully,  and  if  this  is  done,  the  draw- 
ing of  the  remainder  of  the  problem  will  be  comparatively 
easy.  The  interedge  lines  are  determined  in  the  same  man- 
ner and  are  shown  on  the  plate  by  similar  letters.  The 
method  of  procedure  is  precisely  the  same  as  already 
explained,  the  resulting  developments  at  {I))  and  (c)  com- 
pleting the  problem. 

In  drawings  of  this  class,  the  student  will  perceive  that 
the  true  length  of  the  parallel  lines  is  found  by  projection 
drawing.  The  application  of  the  rules  found  under  the 
heading  "  General  Rules  for  Obtaining  Parallel  Develop- 
ments"  is  then  sufficient  for  the  development  of  any  solid; 
and  if  this  application  is  carefully  made,  the  student  will 
meet  with  no  obstacles  that  may  not  be  readily  overcome. 

30.  Development  of  Intricate  Solids. — In  the  process 
of  developing  intricately  formed  solids,  it  becomes  neces- 
sary to  exercise    diligence   and    caution    in  the  observance 


JUNE  25J893. 


Cop>-r:ght,  1S99,  by  THE  Co 
All  right 


D 

n 

n 

r 

D 
•0 

n 


in 


KV  Engineer  Company. 
erved. 


JOA/JV  SM/TH,  CLASS  N9  4529. 


§  10      DEVELOPMENT  OF  SURFACES       33 

of  the  regular  order  of  unrolling"  the  surfaces  on  the  stretch- 
out and  carefully  to  draw  the  developers  to  their  corre- 
sponding edge  lines  in  the  development.  Note  that  in 
order  to  obtain  the  outline  of  the  development  for  the  inter- 
secting solid  at  Fig.  2  (c),  a  separate  developer  is  required 
in  the  case  of  each  edge  or  interedge  of  the  solid. 


DRAWI:N"G    PLiATE,  TITI^E:    DEVELOPMEIS^TS    III 


PROBLEM  7 

31.     To  (leveloi)  cylinders  tliat  intersect  irregularly. 

This  problem  is  a  development  of  the  cylinders  projected 
in  Practical  Projection  to  show  views  of  intersecting  cylin- 
ders of  unequal  diameters. 

Construction. — The  principles  are  the  same  as  those 
governing  previous  developments,  but  since  many  lines  are 
required  for  the  drawing,  the  student  should  carefully  fol- 
low the  directions  in  the  order  stated.  Draw,  first,  the 
projections  as  at  Fig.  1  {a)  of  this  plate  and  carefully 
indicate  the  line  of  intersection  of  the  two  solids.  The 
development  of  the  smaller  cylinder  is  then  made.  Space 
the  outline  of  the  full  view  as  at  (/;)  and  (/>'),  using  10  spaces; 
number  these  points  in  a  corresponding  manner,  as  shoAvn, 
and  project  them  to  their  adjacent  views.  Designate  the 
intersections  in  the  different  views  by  similar  numbers,  to 
avoid  confusion.  Develop  the  stretchout  J/ iV  in  the  usual 
way;  draw  edge  lines  and,  afterwards,  developers,  as  shown 
on  the  plate.  The  development  at  (r)  is  thus  completed, 
the  irregular  outline  there  shown  being  the  pattern  for  the 
surface  of  the  smaller  cylinder.  This  part  of  the  work  may 
be  completed  without  reference  to  the  larger  cylinder.  In 
fact,  it  may  be  considered  as  a  separate  drawing,  and  the 
student  should  take  no  notice  of  any  other  lines  on  the 
drawing  except  those  pertaining  to  the  smaller  solid.  In 
this  way  he  will  accustom  himself  to  working  on  drawings 
that  overlap  one  another. 


34       DEVELOPMENT  OF  SURFACES      §  16 

A  saving  of  the  draftsman's  time  is  often  effected  by 
thus  making  developments  over  other  portions  of  the  draw- 
ing, for  if  this  were  not  done,  a  separate  projection  would 
be  required  for  each  development.  The  large  number  of 
lines  on  the  drawing  is,  however,  apt  to  be  confusing  to  the 
beginner,  and  is  frequently  a  cause  of  error  when  the  draw- 
ing is  transferred  to  the  metal.  Unusual  care  must  there- 
fore be  taken  in  such  cases. 

The  development  of  the  smaller  cylinder  being  completed, 
attention  is  directed  to  the  larger  solid.  It  is  evident  that 
the  outline  of  the  development  of  the  larger  cylinder  will  be 
a  parallelogram  having  an  irregularly  outlined  portion  in 
some  part  of  the  figure.  Reference  to  the  plan  indicates 
that  the  edges  assumed  for  the  smaller  cylinder  intersect  the 
surface  of  the  larger  at  9,  10,  11,  12,  13,  IJ^,  15,  16,  and  1, 
the  total  distance  from  .9  to  1  being  one-quarter  of  the 
length  of  its  outline.  These  points  may  now  be  assumed  as 
the  edges  for  that  portion  of  the  surface  of  the  larger  cylin- 
der, the  remainder  being  spaced  off  in  the  usual  manner,  as 
A,  B,  C,  etc.  Any  convenient  point  on  the  outline  may  be 
taken  as  the  starting  place  for  the  stretchout,  the  extreme 
upper  point  of  the  plan  at  A  being  selected  in  this  case. 
Since  the  intersections  occur  only  on  the  assumed  edges  that 
are  indicated  by  the  numbered  points,  the  drawing  of  the 
edge  lines  in  the  development  may  be  omitted  through  the 
lettered  points,  the  outer  edge  lines  alone  being  necessary  to 
define  the  size  of  the  parallelogram.  Developers  are  then 
drawn  from  the  same  points  of  intersection  in  the  elevation 
used  for  the  development  of  the  smaller  cylinder,  care  being 
exercised  that  they  extend  to  their  corresponding  edge  lines 
in  the  development  of  the  larger  cylinder,  as  shown  on  the 
plate  at  Fig.  1  {d).  * 


*  It  will  be  noted  during  the  drawing  of  this  development  that  the 
stretchouts  of  these  cylinders  are  measured  as  chord  distances.  The 
lengths  in  {d)  for  the  large  cylinder,  therefore,  will  be  slightly  different 
as  measured  from  a  to  1  when  comparison  is  made  with  the  length  1  to 
9,  and,  correspondingly,  from  A  to  H.  The  student  should  understand 
that  in  actual  shop  practice  these  chord  distances  are  equally  spaced  in 
all  quadrants  in  order  to  overcome  this  difficulty.  The  difference  is, 
however,  so  slight  in  this  drawing  that  no  allowance  is  necessary. 


§  1(3  DEVELOPMENT  OF  SURFACES  35 

PIIOIJI^EM  8 

32.  To  (levelop  an  octagonal  prism  and  an  intersect- 
ing' cylinder. 

Explanation. — The  projections  given  \\\  Practical  Projec- 
tion to  show  views  of  a  prism  intersected  by  a  cylinder  are 
redrawn  for  this  problem.  The  development  of  the  cylinder 
is  very  similar  to  that  of  the  smaller  cylinder  in  the  prece- 
ding- problem,  as  shown  at  Fig.  2  {c).  Note  that  the  edges  / 
and  .9,  assumed  on  the  surface  of  the  cylinder,  coincide  with 
the  edge  C  of  the  prism  ;  this  simplifies  the  development, 
and  the  student  will  have  no  difficulty  in  completing  the 
work  at  (r),  it  being  similar  to  that  described  in  the  preceding 
problem. 

Construction.— Since  the  development  of  one-half  of  the 
prism  will  serve  to  illustrate  the  principles  of  this  problem, 
four  sides  only  need  be  laid  out  on  the  stretchout.  In  this 
case,  the  edge  lines  are  to  be  drawn  and  indicated  as  such, 
in  order  that  the  distinction  between  them  and  the  inter- 
edge  lines  may  be  clearly  marked,  in  accordance  with  the 
instructions  given  under  the  heading '"  Development  by 
Parallel  Lines."  After  developing  the  stretchout  as  shown 
at  (<'/),  the  completion  of  the  development  is  made  in  the 
same  manner  as  heretofore. 


PROIJLEM  9 

33.  To  develop  the  surfaces  of  a  cylinder  inter- 
secting a  SI  here. 

Explanation.— The  projections  of  the  sphere  intersected 
by  a  cylinder  that  were  drawn  for  Practical  Projection  are 
here  reproduced  at  Fig.  3  {a)  on  the  plate,  but  to  a  smaller 
scale.  This  problem  introduces  no  new  element  in  the 
development  of  solids  by  parallel  lines.  It  is  given  a  place 
in  this  instruction  merely  to  show  the  student  that,  while 
the  task  of  finding  the  line  of  intersection  of  parallel-lined 
solids  with  solids  of  other  classes  may  depend  on  various 
principles  of  projection  drawing,  their  development  after  this 


36       DEVELOPMENT  OF  SURFACES      §  16 

line  has  been  produced  is  the  same  in  all  cases.  After  the 
completion  of  the  projections,  the  outline  of  the  cylinder,  as 
shown  in  the  plan,  is  divided  into  spaces  by  locating  a  con- 
venient number  of  points  ;  this  number  is  always  optional 
with  the  draftsman,  it  being  customary  to  take  as  many  as 
are  required  for  accurate  development  ;  for,  of  course,  the 
more  points  there  are,  the  greater  will  be  the  accuracy. 

CoxsTRUCTiox. — In  this  case,  as  in  the  case  of  all  symmet- 
rical solids,  it  is  necessary  to  locate  points  on  but  one-half  of 
the  outline,  since  their  projectors,  if  produced,  will  fix  cor- 
responding points  on  the  remaining  portion  of  the  outline. 
The  assumed  edges  are  then  projected  to  the  elevation  and 
their  intersections  with  the  sphere  carried  by  developers  to 
the  development  at  {b).  A  similar  development  may  be 
completed  at  {b')  by  producing  the  edge  lines  to  that  figure 
and  drawing  developers  as  shown,  completing  the  problem. 

The  stretchout  line  has  been  prominently  used  in  these 
problems.  In  many  cases  it  may  be  convenient  to  develop 
the  stretchout  on  one  of  the  lines  used  for  developers,  thus 
avoiding  an  additional  line  on  the  drawing.  Generally,  in 
sheet-metal  work,  it  will  be  found  preferable  to  give  the 
stretchout  line  the  same  prominence  as  in  these  problems, 
for  reasons  that  will  be  apparent  in  the  studentls  later  work. 


DEVEIiOPlMEXT    BY    RADIAL    UXES 

34.      Relatiou    of    tlie    Pyrauiid    to    tlie    Cone. — We 

now  come  to  the  second  general  division  of  those  solids 
whose  development  may  be  accurately  accomplished — that 
is,  solids  developed  on  radial  liins.  It  is  here  proposed  to 
show  the  relation  borne  by  the  pyramid  to  the  cone,  since 
the  methods  of  development  of  both  solids  are  similar.  In 
the  development  of  the  cylinder,  its  curved  outline  was 
divided  by  points  into  a  number  of  equal  parts.  These 
points  in  the  plan  were  then  projected  to  the  elevation  and 
there  considered  as  assumed  edges;  in  other  words,  the 
cvlinder  was  treated  as  a  many-sided  polygonal  solid,  this 


§16 


DEVELOPMENT  OF  SURFACES 


37 


solid  appearing  inscribed  within  the  cylinder.  In  a  similar 
manner  we  shall  now  consider  the  pyramid  as  an  inscribed 
solid  in  its  relation  to  the  cone.  This  may  be  better  under- 
stood by  reference  to  Fig.  23  (a),  (/;),  and  (c). 


(bj 


The  removal  from  the  cone  at  {(7)  of  the  pieces  marked  x 
leaves  a  solid  that  may  be  recognized  as  a  quadrangular 
pyramid — better  shown  at  (d),  the  pieces  being  removed 
and  shown  in  an  adjacent   position.      The  illustration  at  (c) 


38 


DEVELOPMENT  OF  SURFACES 


16 


is  a  perspective  elevation  of  the  parts  shown  at  {a)  and  (Z?), 
and  is  introduced  for  the  purpose  of  showing  the  relation 
between  the  cone  and  the  pyramid  to  better  advantage. 
Comparing  these  two  solids  in  the  figure,  it  will  be  seen  that 
the  edges  of  the  pyramid  at  (d)  correspond  to  certain  elements 
of  the  cone  at  (a).  Further,  it  will  be  seen  that,  if  the  cone 
is  covered  with  paper,  as  in  Fig.  24,  and  the  position  of  the 
elements  noted,  the  paper  afterwards  being  unrolled  as 
shown,  the  elements  may  be  imagined  as  leaving  their 
imprint  on  the  paper,  as  at  o  a,  o  b,  etc.,  Fig.  24.  The  posi- 
tion of  these  imprints  will  correspond  relatively  to  the  loca- 
tion of  the  elements  on  the  surface  of  the  solid.  A  figure 
similar  to  the  unrolled  covering  may  be  described  by  the  aid 


Fig.  24 

of  the  compasses,  the  boundaries  of  the  development  being 
defined  by  the  position  of  any  element  chosen  at  pleasure, 
notice  being  taken  of  the  first  and  last  contact  of  such  ele- 
ment with  the  drawing  during  a  single  revolution  of  the 
cone.  The  radius  of  the  arc  thus  described  is  always  equal 
to  the  true  length  of  an  element,  while  its  length  is  equal  to 
the  circumference  of  the  base  of  the  cone. 

Suppose,  now,  that  the  surface  of  the  pyramid  shown  at 
Fig.  23  {b)  is  covered  in  a  similar  manner,  the  covering  being 
afterwards  tmrolled  as  shown  in  Fig.  25.  It  will  be  found 
that  an  arc  described  with  a  radius  equal  to  th*at  used  for 
the  development  of  the  cone  in  Fig.  24 — that  is,  equal  to  the 
true  length  of  an  edge  of  the  pyramid — will  pass  through 
the  points  a,  b,  c,  d,  and  a\  representing  the  lower  extremi- 
ties of  the  upright  edges  of  the  pyramid.     These  points  are 


§10 


DEVELOPMENT  OF  SURFACES 


39 


equally  distant  from  one  another  as  measured  on  the  arc. 
This  may  be  proved  by  setting  the  dividers  at  a  distance 
equal  to  the  length  of  the  base  edge  of  the  pyramid  and 
stepping  off  the  spaces  on  the  arc.  The  only  difference 
between  the  developments  for  these  two  solids  lies  in  the  fact 


Fig.  25 

that,  for  the  cone^  the  development  is  defined  by  the  circular 
arc,  while  in  the  case  of  the  pyramid,  straight  lines-  are 
drawn  between  the  points  a.,  b,  c,  etc.,  as  shown  in  Fig.  25. 
These  developments  will  be  completed  in  a  later  problem. 

35.  Stretchouts  for  Radial  Solids.  —  Since  the  dis- 
tance around  the  bases  of  the  cone  and  the  pyramid  may 
be  measured  on  an  arc  whose  radius  is  equal  to  the  length 
of  one  of  the  elements  of  the  cone  (or,  in  the  case  of  the 
pyramid,  to  the  length  of  one  of  its  upright  edges),  such 
an  arc  may  be  described  for  the  stretchout  of  these  solids. 
The  measurement  around  the  solid  being  taken  on  a  partic- 
ular line,  or  base  plane,  it  may  here  be  observed  that  any 
real  or  assumed  base  plane  of  a  cone  or  pyramid  may  be 
treated  in  a  similar  manner.  The  length  of  the  radius  by 
which  the  stretchout  is  described  must  in  all  cases  be 
equal  to  the  true  length  of  the  elements  in  that  portion  of 
the  solid. 

This  is  illustrated  in  Fig.  20,  the  cone  O  A  B  being  devel- 
oped along  a  stretchout  described  with  the  radius  OB.  An 
assumed  base  may  be  taken  at  CD;  an  arc  is  then  described 
from  the  center  O^'ith.  the  radius  O  D  {O  D  being  the  true 
length  of  the  elements  of  the  cone  O  CD).  If  the  width 
of  a  space  between  the  elements  on  the  assumed  base  CD 


40 


DEVELOPMENT  OF  SURFACES 


§16 


(measured  on  the  plan)  is  taken  in  the  dividers  and  spaced 
off  on  the  arc  D  F,  the  spaces  will  be  found  to  coincide  with 
the  intersection  of  the  elements  in  the  development  drawn 
from  the  arc  B  E.  The  same  will  be  found  true  for  any 
right  base  that  may  be  assumed  for  the  cone.  As  a  matter  of 
precaution,  it  is  customary,  when  drawing  developments  of 

the  radial  solids,  to  de- 
scribe the  stretchout 
with  as  long  a  radius 
as  possible,  usually  not 
exceeding  the  length  of 
the  elements  or  edges 
shown  in  the  draw- 
ing. For  reasons  that 
will  be  shown  during 
the  construction  of  the 
problems,  it  is  con- 
venient to  locate  the 
center  for  describing 
the  stretchout  at  the 
vertex  of  the  solid. 

36.     Ee  volution 
of     Radial     Solids. — 

During  the  study  of 
projection  draAving,  it 
was  learned  that  meas- 
urements for  all  dis- 
tances and  the  posi- 
tion of  points  on  the 
surfaces  of  cones  and  pyramids — radial  solids — are  deter- 
mined by  means  of  their  radial  lines,  or  elements.  The  same 
principle  must  be  adhered  to  Avhen  developments  of  such 
solids  are  produced.  It  is  essential,  therefore,  that  these 
lines  should  be  shown  in  their  true  lengths  on  drawings  from 
which  such  developments  are  produced.  This  will  necessi- 
tate much  work  on  the  part  of  the  draftsman,  especially  if 
the  surfaces  are  in  any  way  irregular  or  are  intersected  by 


Fig.  26 


§16 


DEVELOPMENT  OF  SURFACES 


41 


other  solids.  To  overcome  the  necessity  for  drawing  a 
number  of  views,  advantage  is  taken  of  a  principle  that 
may  be  observed  during  the  revolution  of  the  solid.  This 
revolution  is  effected  in  a  very  simple  way  on  the  drawing 
board,  and  an  illustration  of  the  method  used  is  furnished 
in  Fig.  27.  The  student  will  understand,  from  an  inspec- 
tion of  that  figure,  that  if  the  cone  O  A  B  is,  revolved  on 
its  axis  in  the  direction  of  the  arrow,  the  motion  of  any 
point  on  its  surface  Avill  be  indicated  on  the  elevation  by 
a  horizontal  line.  When  any  elements  of  the  cone  are  in 
the  positions  occupied  by  the  elements  O  A  and  O  B,  their 
true  lengths  are  shown  on  the  elevation,  and  measurements 
may,  therefore,  be  taken  from  such  lines  or  from  any  points 
located  on  these  lines.  Lines  in  this  position  may  be  called 
true  edge  li)ics,  their  use  under  this  name  being  peculiar  to 
the  radial  solids. 

When  developing  the  surfaces  of  pyramids  shown  in  cer- 
tain positions,  it  is  sometimes  necessary  to  draw  this  true 
edge  line  independently  of  the  figure.  A  problem  illustra- 
ting this  principle  is  given  on  the  following  plate.  Since  the 
elements  of  a  cone  are  equal  in  length, 
the  position  of  any  point  that  may  be 
located  on  any  of  these  elements  (as  x, 
located  on  the  element  O  C,  Fig.  27) 
may  be  projected  in  a  horizontal  direc- 
tion to  the  true  edge  line  at  x",  and  its 
correct  distance  from  the  vertex  O  be 
there  ascertained.  The  point  x,  there- 
fore, could  be  projected  to  either  ele- 
ment O  A  or  OB,  but  the  supposed  rev- 
olution is  generally  represented  as  being 
made  towards  the  true  edge  line  that  is 
nearest  the  point  whose  location  it  is  de- 
sired to  determine,  although  in  Fig.  27 
the  point  x  is  moved  in  the  opposite 
direction.  Thus,  in  Fig.  27,  if  it  is  de- 
sired to  determine  the  exact  distance  of 
the  point  x  from  the  vertex  (9,  the 
M.  E.    V.-?i 


Fig.  s}7 


4-2  DEVELOPMENT  OF  SURFACES  §  16 

horizontal  line  x.r"  is  drawn  in  the  elevation,  and  the  distance 
from  the  point  O  to -the  point  x'  is  then  the  exact,  or  true, 
distance  between  the  two  points.  The  same  result  would 
be  obtained  if  an  elevation  were  drawn  showing  the  ele- 
ment O  C  in  its  true  length ;  but  as  seen  from  the  foregoing 
explanation,  the  method  here  explained  is  much  shorter  and 
better  adapted  to  the  wants  of  the  draftsman. 

37.  Use  of  Construction  Lines. — In  the  development 
of  radial  solids,  the  same  construction  lines  are  used  as  in 
obtaining  the  developments  for  parallel  solids,  although  in  a 
slightly  different  manner.  Since  the  stretchout  line  has 
been  shown  in  its  adaptation  to  the  development  of  these 
solids,  it  may  be  represented  in  a  similar  manner  on  the 
drawings.  Developers  also  are  indicated  by  the  same  kind 
of  broken  lines  used  in  the  preceding  class,  but,  like  the 
stretchout.  the\^  are  described  in  the  form  of  arcs,  and  must 
be  produced  from  an  element  or  edge  that  is  shown  in  its 
true  length.  In  a  similar  manner,  these  developers  are 
drawn  to  the  development,  arcs  being  described  extending 
from  points  on  edges  or  interedges,  as  the  case  may  be,  to 
their  corresponding  edge  lines  or  interedge  lines  in  the 
development.  When  edge  lines  and  interedge  lines  occur, 
they  converge  to  a  point,  and  are  the  radial  lines  by  which 
this  class  of  solids  is  distinguished.  Such  edges  and  inter- 
edges are  re])resented  by  lines  similar  to  those  used  for  the 
same  purpose  in  parallel  developments  and  refer  to  the  same 
corresponding  portions  of  the  solid. 


GEXERAL  RITLES  FOR  OBTAINTN'G  R-VDIAL 
DEVELOPMENTS 

38.  A  slight  modification  of  the  rules  for  obtaining 
developments  on  parallel  lines  is  here  applied  to  the  develop- 
ment of  radial  solids.  In  connection  with  the  foregoing 
instruction,  a  comparison  of  the  two  sets  of  rules  will 
enable  the  student  to  understand  the  principles  by  which 
these  developments  may  be  accomplished. 


§  16  DEVELOPMENT  OF  SURFACES 


43 


1.  A  projection  must  first  be  drawn,  consisting  of  a  plan 
and  elevation  and  showing  the  solid  in  a  right  position. 

•2.  The  development  is  always  obtained  from  that  view  in 
which  the  axis  of  the  solid  is  shown  in  its  true  length  (since 
the  revolution  of  the  solid  may  not  readily  be  shown  in  any 
other  view). 

3.  The  stretchout  is  described  with  a  radius  equal  to  the 
length  of  the  true  edge  of  the  solid.  Its  center  may  be 
conveniently  located  at  the  vertex. 

4.  To  indicate  the  width  and  relative  position  of  the  sur- 
faces, points  are  located  on  the  stretchout  corresponding  to 
the  position  of  those  points  on  the  outline  of  a  sectional  or 
base  view.  This  view  must  be  taken  at  right  angles  to  the 
axis  of  the  solid,  the  distance  from  the  vertex  being  deter- 
mined by  the  length  of  the  true  edge  lines  in  the  elevation. 

5.  Edge  lines  and  interedge  lines  are  always  radii  of  the 
stretchout  arc. 

G.  Points  located  on  the  surface  of  the  solid  must  be  pro- 
jected to  the  true  edge  line  by  projectors  drawn  at  right 
angles  to  the  axis. 

7.  Developers  are  described  Avith  radii  equal  to  the  dis- 
tances on  a  true  edge  line  from  the  vertex  to  the  points  pro- 
jected to  such  edge  line.  Each  developer  extends  thence 
to  its  corresponding  edge  line  or  interedge  line  in  the 
development. 

8.  Interedge  lines,  when  necessary  for  the  development, 
must  be  indicated  on  the  projection  as  well  as  on  the  develop- 
ment. Points  located  on  such  lines  are  projected  to  the 
true  edge  lines  and  thence  developed  in  the  usual  manner. 

9.  The  lengths  of  the  outer  edge  lines  in  a  complete 
development  of  a  solid  must  be  defined  by  the  same 
developers. 

The  application  of  these  rules  will  be  made  apparent  to 
the  student  in  the  construction  of  the  following  problems, 
which  involve  the  development  of  radial  solids. 


44  DEVELOPMENT  OF  SURFACES  §  16 

DRAWIXG    PliATE,   TITLE  :     DEVELOPMEXTS  IT 


PROBLEM   10 

39.     To  develop  tlie  surface  of  a  cone. 

Construction. — Draw  a  plan  and  an  elevation  of  the  cone, 
as  shown  at  Fig.  1  on  the  plate.  The  cone  is  there  shown 
in  a  right  position,  the  dimensions  being  1^  inches  in  diam- 
eter at  the  base  and  2  inches  high.  The  true  length  of  its 
axis  is  shown  in  the  elevation,  and  the  development  is  there- 
fore made  from  that  view.  Divide  the  outline  of  the  base  in 
the  plan  into  a  convenient  number  of  equal  parts  (in  this 
case  12) ;  from  the  vertex  O'  of  the  cone  as  a  center,  describe 
the  stretchout  arc  B'  a'  with  a  radius  equal  to  the  true  length 
of  the  elements  of  the  cone  (that  is,  the  distance  O'  B'  in 
the  elevation).  With  the  dividers,  take  the  length  of  one  of 
the  equal  spaces  in  the  plan,  and,  starting  at  a  convenient 
point  on  the  stretchout,  as  at  «,  step  off  spaces  equal  in 
number  to  those  on  the  plan,  thereby  making  the  length  of 
the  stretchout  equal  to  the  circumference  of  the  base  of  the 
cone.  From  each  of  the  points  thus  located  on  the  stretch- 
out, an  edge  line  may  be  drawn  to  the  vertex  O' ;  but  since 
there  are  no  points  on  the  surface  of  the  cone  that  it  is  desir- 
able to  locate  in  this  instance,  only  the  outer  edge  lines  a  0' 
and  a'  O'  need  be  inked  in  on  the  drawing.  These  lines  are 
to  be  further  indicated  by  means  of  the  small  arrowheads 
(as  in  the  case  of  parallel  solids)  illustrated  on  the  plate 
in  Fig.  1.     This  completes  the  development. 


PROBLEM   11 

40.  To  develop  tlie  surfaces  of  a  quadrangular 
pyramid. 

Construction. — This  development  is  shown  on  the  plate 
at  Fig.  2,  the  right  plan  and  elevation  being  first  drawn 
according  to  the  dimensions  given  in  the  figure.  It  will 
be  seen  that  the  true  length  of  the  edge  is  shown  in  the 


J^AE  25,  J 89 3. 


Copyright.  1899,  by  The  Cc, 
All  right 


Ekv  Engineer  Company. 
served. 


JOHN  SM/TH.  CLASS  N?  4529. 


§  16      DEVELOPMENT  OF  SURFACES       45 

elevation;  the  stretchout  may,  therefore,  be  described  as  in 
the  case  of  the  cone  in  the  preceding  problem.  After  set- 
ting the  dividers  to  the  width  of  one  of  the  base  edges  shown 
in  the  plan  at  A  B,  Fig.  2,  begin  at  a  and  step  off  on  the 
stretchout  line  spaces  equal  in  number  to  the  base  edges 
of  the  pyramid.  Thus,  points  are  located  at  a,  d,  c,  d,  and  a' . 
Draw  lines  connecting  these  points  in  the  manner  shown, 
and  draw  other  lines  from  each  of  these  points  to  the 
vertex  O' .  In  this  case  the  edge  lines  must  be  drawn  in 
the  development,  since  there  are  actual  edges  on  the  solid; 
besides,  it  is  necessary  to  define  those  portions  of  the  devel- 
opment as  indicated  on  the  plate.  Complete  the  drawing 
in  the  manner  shown,  the  outline  O'  a  b  c  d  a'  being  the 
development  of  the  pyramid. 


PROBLEM  13 

41,  To  develop  tlie  surfaces  of  an  octagonal  pyra- 
mid. 

Explanation. — The  base  of  the  pyramid  whose  dimen- 
sions are  given  in  Fig.  3  on  the  plate  is  not  a  true  octagon, 
the  alternate  sides  only  being  equal.  It  will  be  seen, 
however,  that  the  octagon  may  be  circumscribed  by  a 
circle;  that  is,  a  circle  may  be  described  in  the  plan 
from  the  center  (9  with  a  radius  O  A^  whose  outline  will 
pass  through  all  the  points  A,  B,  C,  D,  etc. ;  therefore, 
tTie  development  of  the  solid  may  be  accomplished  by  this 
method. 

Construction. — The  true  length  of  the  edge  lines  is  not 
shown  in  either  view  presented,  and  it  is  therefore  necessary 
to  draw  a  line  that  will  represent  the  true  edge  in  the  eleva- 
tion. This  is  found  as  follows:  From  (9  as  a  center,  with 
the  radius  OH,  describe  the  arc  H  H',  intersecting  a 
line  OH'  (drawn  parallel  to  the  base  line  of  the  front  eleva- 
tion) at  H'\  project  the  point  H'  to  the  base  line  of  the 
front  elevation  (extended)  at  H"  \  (9' //"  is  then  the  true 
length  of  an  edge  of  the  pyramid,  and  is,  at  the  same  time. 


46       DEVELOPMENT  OF  SURFACES      §  16 

the  length  of  the  stretchout  radius.  Next,  describe  the 
stretchout  with  this  radius  from  the  vertex  O'  as  a  center; 
then  space  off  the  width  of  the  surfaces  shown  on  the  base 
in  the  plan,  as  at  a,  h.  g,  /",  etc.,  and  complete  the  develop- 
ment as  directed  in  the  preceding  problem. 


PROBLEM    13 

43.  To  develoj)  tlie  surfaces  of  au  irregular  frus- 
tum of  a  liexagoual  ijyraniid. 

Construction. — The  projections  shown  in  Fig.  4  on  the 
plate  are  first  drawn  in  accordance  with  the  dimensions 
there  given,  thus  producing  a  right  plan  and  elevation  of 
the  frustum.  In  this  and  all  similar  developments,  it  is 
desirable  to  extend  the  edges  of  the  p5^ramid  to  the  vertex 
of  the  solid.  Since  the  drawing  does  not  show  the  true 
length  of  the  edge,  this  must  first  be  found  by  the  method 
described  in  the  preceding  problem,  and  produced  as  shown 
in  the  elevation  at  O'  a.  The  points  in  the  upper  portion  of 
the  solid,  at  B\  C\  D\  are  then  projected  to  the  true  edge 
line  at  B'\  C" ,  and  D" .  The  stretchout  is  next  described 
from  O'  as  a  center  with  a  radius  O'  a ;  the  widths  of  the  sur- 
faces are  then  laid  off  at  a,  b,  r,  d,  etc.,  and  the  corresponding 
edge  lines  are  then  drawn.  Developers  are  now  described 
from  B'\  C",  and  D" ,  as  previously  directed,  the  intersec- 
tions with  their  corresponding  edge  lines  being  noted  at 
b'  a',  c'  f\  and  d'  c' .  Complete  the  development  by  drawing 
its  full  outline  and  adding  the  indicators  in  the  manner 
shown. 


PROBLEM    14 

43.     To  develop  an  irregular  frustum  of  a  cone. 

Two  developments  are  required  for  this  problem,  one  of 
them  being  fully  described  and  the  other  to  be  drawn  by  the 
student  as  a  test  of  his  advancement. 


§  16      DEVELOPMENT  OF  SURFACES   '    47 

Construction.— The  projection  drawings  for  the  first 
development  are  shown  on  the  plate  at  Fig.  5,  that  portion 
of  the  cone  representing  a  parabolic  section  being  presented 
for  development.  In  this,  as  in  the  preceding  case,  the 
completion  of  the  figure,  as  shown  by  the  broken  lines,  must 
first  be  made.  Proceed  then  as  if  the  complete  cone  were 
to  be  developed;  that  is,  divide  the  outline  of  the  base  in 
the  plan  into  a  number  of  equal  spaces,  observing  that  cer- 
tain of  the  points  fall  on  the  ends  of  the  parabolic  curve  (as 
at  Fand  B  in  the  plan.  Fig.  5).  The  elements  of  the  cone 
are  then  produced  in  the  elevation,  their  intersections  with 
the  edge  of  the  frustum  at  b'\  c" ,  d'\  etc.,  being  then  pro- 
jected as  before  to  the  true  edge  line.  As  in  former  prob- 
lems, the  stretchout  is  next,  described  and  spaced  off,  and 
edge  lines  are  drawn.  Developers  may  now  be  described 
as  shown  on  the  plate  and  the  development  completed  in 
the  usual  manner.  The  irregular  curve  is  now  traced 
through  the  points  thus  determined.  The  projection  draw- 
ings for  the  second  development  of  this  problem  are  shown 
on  the  plate  at  Fig.  G,  and,  since  the  methods  used  are  the 
same  as  in  the  case  just  described,  the  development  may  be 
completed  by  the  student  without  further  instruction. 


PROBLEM    15 

44.     To  develop  the  surfaces  of  Intei-secting  cones. 

The  projections  for  this  problem  are  shown  on  che  plate 
at  Fig.  7.  They  should  first  be  carefully  drawn  by  the 
student,  the  line  of  intefsection  being  accurately  determined 
by  the  method  for  finding  the  intersection  of  two  cones  ^that 
was  given  in  Practical  Projection.  After  this  line  has  been 
found,  all  construction  lines  used  in  the  projection  should 
be  erased  from  the  drawing;  if  this  is  not  done,  confusion 
is  liable  to  result. 

Construction.— The  development  of  these  surfaces  does 
not  differ  materially  from  those  in  the  preceding  problem, 
as  an  inspection  of  Fig.  7  will  indicate.  First  draw  the 
development  of  the  surface  of  the  smaller  cone;  extend  the 


48       DEVELOPMENT  OF  SURFACES      §  16 

edge  lines  in  the  elevation  to  the  vertex  0\  as  shown; 
extend  them  also  to  the  assumed  base  a  e  and  produce  the 
full  view  of  the  base  in  the  manner  indicated  by  the  broken 
lines  at  («),  using  only  one-half  of  the  circumference.  The 
semi-circumference  is  then  spaced  off  as  at  a,  b,  c,  d,  and  e\ 
b,  c,  and  d  slvq  then  projected  to  the  base  line  a  e;  and  the 
elements  b'  O' ,  c'  O',  and  d'  O'  are  drawn  as  shown.  The 
intersections  of  these  elements  with  the  line  of  intersection 
of  the  two  solids  are  then  projected  to  the  true  edge  of  the 
smaller  cone,  at  a",  b" ,  etc.  A  stretchout  for  this  solid  is  next 
described  from  the  vertex  O'  as  a  center  with  a  radius  O'  e; 
the  points  a,  b,  c,  etc.  at  (b)  are  located  by  spacing  the 
stretchout;  and  the  distances  a  b,  be,  etc.  are  taken  from 
the  full  view  at  («).  Next,  draw  the  edge  lines  a  O',  b  0\  c  O', 
etc.  at  (b) ;  and  from  points  a",  b",  etc.  describe  developers 
in  the  manner  shown.  The  irregular  curve  at  {b)  is  then 
traced,  completing  the  development  for  this  solid. 

As  a  matter  of  convenience  in  this  case,  the  development 
of  the  larger  cone,  or,  rather,  as  much  of  its  surface  as  will 
show  the  opening  made  by  the  intersection  of  the  smaller 
cone,  may  be  drawn  to  the  right  of  the  projections,  as  at  {c). 
Now  draw  the  horizontal  center  line  in  the  plan  and  from 
the  center  of  the  plan  draw  a  line  tangent  to  the  line  of 
intersection  of  the  cones,  marking  the  point  where  this  line 
meets  the  base  4-'-  Divide  l'-4-'  into  a  convenient  number 
of  parts  (in  this  case  3)  and  project  the  points  1',  2',  3',  4'  to 
the  elevation.  Draw  elements  from  O"  to  the  projected  points 
and  mark  the  points  where  these  elements  cross  the  line 
of  intersection  of  the  cones  1,  S,  3,  4,  5,  6,  and  7.  The 
points  /,  2',  3',  4'  in  the  plan  establish  the  width  of  the 
spaces  that  shall  be  stepped  off  on  the  stretchout.  Describe 
the  stretchout  from  the  center  O",  as  shown  at  (c),  and  on 
this  line  set  off  the  spaces  determined  in  the  plan  at  1',  2\  3', 
and  4',  and  repeat  them  on  both  sides  of  the  edge  line  1  O", 
as  indicated  at  (c)  Project  the  points  J,  2,  3,  etc.  to  the 
right-hand  true  edge  line  of  the  elevation  and  carry  them 
thence,  by  developers,  to  the  drawing  at  (c),  completing  the 
development  as  there  shown. 


§  16     DEVELOPMENT  OF  SURFACES       49 

The  student  will  readily  perceive  that  if  the  line  of  inter- 
section of  the  two  cones  is  not  correctly  drawn,  the  true 
development  cannot  be  produced,  since  the  drawing  depends 
entirely  on  the  line  thus  determined.  If  these  points  are 
incorrectly  located,  the  development  is  necessarily  wrong. 
The  importance  of  having  a  correct  projection  is  thus  evident. 

45.  How  to  Recognize  Radial  Forms.— Since  many 
objects  of  frequent  occurrence  in  the  trades  are  portions  of 
the  cone  or  pyramid,  the  student  should  familiarize  himself 
with  the  appearance  of  such  frustums.  Many  objects  rep- 
resent a  combination  of  the  surfaces  of  cones  of  unequal 
sizes,  and  at  first  sight  it  might  appear  that  their  develop- 
ment should  be  produced  only  by  triangulation.  But  an 
experience  in  the  projection  of  cones  and  conic  sections  will 
often  enable  the  student  to  refer  such  sections  to  the  appro- 
priate form,  in  many  cases  thereby  avoiding  a  tedious  proc- 
ess of  development. 

The  representation  of  the  elements  of  the  cone  in  both 
plan  and  elevation  will  determine  the  method  to  be  used  in 
the  development  of  a  given  solid.  In  a  certain  case,  a  num- 
ber of  vertexes  may  be  defined;  each  cone  is  then  to  be 
traced  out  carefully  and  the  development  of  its  surface  made, 
or  as  much  of  that  surface  as  is  required  for  the  object  in 
view,  the  operation  being  similar  to  that  of  the  preceding 
problems. 


TRIANGIJIiATIO]^^ 

46.     Elementary     Principles.  —  Triangulation  —  the 

process  by  which  the  development  of  solids  belonging  to  the 
third  general  division  is  accomplished— is  generally  regarded 
by  sheet-metal  workers  as  particularly  intricate  and  difficult. 
It  is,  on  the  contrary,  a  very  simple  method  of  develop- 
ment, and  should,  to  the  student  that  thoroughly  under- 
stands the  principles  of  projection,  present  no  serious 
obstacles.  This  process  depends  for  its  results  on  two  gen- 
eral principles,  both  of  which  have  been  mentioned  in  this 


50       DEVELOPMENT  OF  SURFACES      ^  10 

section:  first,  to  find  the  true  length  of  all  lines,  real  or 
assumed,  appearing  on  the  surfaces  of  the  solid;  second, 
having  determined  the  true  length  of  such  lines,  to  con- 
struct triangles  similar  in  form  and  relation  to  those  shown 
on  the  solid. 

Certainly,  the  construction  of  a  triangle  whose  three  sides 
are  given  is  not  a  difficult  problem ;  and  the  task  of  finding, 
from  the  projection  drawing,  the  true  lengths  of  its  sides 
involves  nothing  but  the  elementary  principles  of  that 
study.  Having  found  the  true  lengths  of  the  sides  of  such 
triangles  as  are  involved  in  a  development,  nothing  appar- 
ently remains  but  to  show  the  method  of  arranging  the 
triangles  in  their  proper  relation  to  one  another  on  the 
different  surfaces  of  the  solids.  Since  this  is  naturally  sug- 
gested by  the  shape  of  the  solid  itself,  several  solids  are 
presented  in  the  accompanying  problems,  from  the  study  of 
which  the  student  should  learn  to  apply  the  principles  to  the 
development  of  any  solid  of  this  class. 

47.  Illiisti*ation  of  Metliocls  Used. — This  method  of 
development  has  been  previously  mentioned,  and  while  this 

principle  of  patterncutting  is 
usually  applied  to  solids  having 
curved  surfaces,  it  is  best  illus- 
trated by  its  application  to  a 
solid  having  plane  surfaces. 
Such  a  solid  is  shown  in  Fig.  38, 
^'*^-  -^  where  a  perspective  view  is  given 

of  what  ma)^  be  termed  a  transition  piece — ^that  is,  a  piece 
used  to  connect  openings  of  different  sizes,  as  in  pipework. 
Both  bases  are  rectangular  and  in  this  case  parallel,  but 
diagonally  arranged  in  their  relation  to  each  other,  as  may 
be  seen  from  the  figure. 

It  is  at  once  seen  that  none  of  the  methods  described 
under  the  headings  of  parallel-  or  radial-lined  solids  will 
apply  to  the  development  of  the  lateral  surfaces  of  this  solid, 
although  it  is  possible  to  project  a  full  view  of  each  of  the 
surfaces  shown  in  the  figure.      Since  this  would  require  an 


S  IG 


DEVELOPMENT  OF  SURFACES 


51 


unusual  amount  of  work  on  the  part  of  the  draftsman,  the 
following  process,  admitting  of  more  rapid  application,  is 
presented. 


48.  Deterniininj?  the  Triangles. — Clearly,  a  repro- 
duction of  these  triangular  surfaces  on  the  flat  surface  of 
the  drawing  board,  in  the  same  corresponding  relation  with 
reference  to  one  another,  will  be  a  development  of  the  solid. 
Apparently,  the  only  difficulty  that  presents  itself  is  the 
fact  that,  in  certain  cases,  the  sides  of  the  triangles  are  not 
shown  in  their  true  lengths.  It  is  necessary  to  determine 
the  true  lengths  of  all  lines,  in  order  that  the  triangles  may 
be  constructed  of  the  same  size  as  they  are  on  the  surfaces 
of  the  solid. 

As  in  all  cases  where  a  development  is  desired,  a  right 
plan  and  elevation  must  first  be  drawn.  This  is  shown  in 
Fig.  29,  and  from  that  illustra- 
tion it  is  seen  that  all  lines  of 
the  solid  that  appear  on  either 
base  are  shown  in  their  true 
lengths.  It  is  therefore  neces- 
sary, before  the  triangles  may 
be  produced,  to  determine  the 
true  lengths  of  the  remaining 
lines  of  the  solid.  As  mentioned 
in  Practical  Projection^  this  is 
most  readily  accomplished  by 
constructing  in  each  case  a 
right-angled  triangle  whose  base 
is  equal  to  the  length  of  any 
foreshortened  line  in  the  plan, 
and  its  altitude  to  the  vertical 
height  of  the  same  line,  as  shown  in  the  elevation.  The 
hypotenuse  of  such  a  triangle  will  then  be  equal  to  the  true 
length  of  the  line.  In  this  case,  the  lines  A  H,  DH,A  E,  B  E, 
etc.,  foreshortened  in  the  plan,  are  all  represented  by  lines 
of  the  same  length.  The  vertical  height  J/,  Fig.  29,  is  the 
same  in  the  case  of  each  line. 


52 


DEVELOPMENT  OF  SURFACES 


§10 


A  single  triangle  constructed  by  the  above  method,  there- 
fore, will  be  sufficient  to  indicate  the  true  length  of  all  lines 
not  shown  in  their  true  length  in  Fig.  29. 
Such  a  triangle  is  constructed  in  Fig.  30; 
the  base  of  the  triangle  A  H,  Fig.  30,  is 
equal  to  the  length  of  A  H,  Fig.  29,  the 
altitude  J/  being  the  same  as  J/,  Fig.  29. 
^  ^   The  hypotenuse  of  this  triangle  is  therefore 

Fig.  30  ^^e  true  lengths  of  the  lines  A  H,  D  H,  etc., 

as  shown  in  Fig.  29.  The  true  lengths  of  all  lines  border- 
ing the  triangular  surfaces  of  the  solid  shown  in  Fig.  29 
having  been  found,  the  triangles  may  now  be  constructed 
on  the  drawing,  care  being  observed  that  the  adjacent 
triangles  are  completed  in  the  same  order  as  they  are  shown 
on  the  solid.  Any  edge  of  the  solid  may  be  assumed  as  a 
starting  place  for  the  operation;  the  true  length  of  such  an 


Fig.  31 

edge  is  then  laid  off  on  a  line  as  at  A  E,  Fig.  31.  The  tri- 
angle A  EB,  Fig.  31,  is  first  constructed;  the  length  of  the 
side  A  E  being  laid  oft",  and  BE,  Fig.  29,  being  of  the  same 
length,  an  arc  may  be  described  in  Fig.  31  from  j5"  as  a  cen- 
ter, with  a  radius  equal  to  EA.  Intersect  this  arc  at  ^, 
Fig.  31,  with  one  described  from  A  as  a  center.  Avith  a 
radius    equal    to  A  B,  Fig.  29.      Draw  A  B  and   E  B,  thus 


JUNE25.Je93. 


-lEKv  Engineer  Company. 
served. 


JOHN  SM/TH,  CLASS  N?  4529. 


§  16      DEVELOPMENT  OF  SURFACES       53 

developing  the  triangle  A  £  JJ,  Fig.  31,  which  is  the  correct 
development  of  the  surface  A  E  B,  Fig.  29. 

The  adjacent  triangle  £  B  F,  Fig.  31,  may  next  be  con- 
structed. Since  B  F  is  equal  to  BE,  Fig.  29,  an  arc  maybe 
described  in  Fig.  31  from  i?  as  a  center,  with  a  radius  BE; 
this  arc  is  then  intersected  by  an  arc  described  from  ^  as  a 
center,  with  a  radius  equal  to  the  length  of  E  F,  Fig.  29, 
thus  developing  the  triangle  E  B  F,  Fig.  31,  which  corre- 
sponds to  the  surface  E  B  F,  Fig.  29.  In  like  manner,  each 
surface  of  the  solid  is  developed,  due  care  being  observed 
that  adjacent  triangles  are  placed  in  corresponding  positions 
in  the  development. 

49.  Completion  of  the  Drawing. — The  completion 
of  the  drawing  is  made  in  a  manner  somewhat  similar  to 
parallel  and  radial  developments:  that  is,  the  edge  lines  may 
be  indicated  as  in  those  methods,  the  outer  edges  being 
denoted  by  full  lines  and  those  lines  on  which  bends  are  to  be 
made  when  the  flat  surfaces  are  formed  up  being  designated 
by  the  customary  indicator  circles.  It  will  thus  be  seen 
that  no  new  principles  are  required  to  produce  developments 
by  this  method.  A  careful  observance  of  the  different  por- 
tions of  the  drawing  is  required,  since  an  object  as  simple 
as  that  shown  in  Fig.  29  is  seldom  met  in  practice.  The 
same  methods  are  used,  however,  and  should  be  readily 
understood  by  the  student  and  applied  to  the  drawing  in  the 
same  manner  as  has  been  shown. 


DRAWING  PliATE,   TITLE:     DEVEIjOPME:N^TS  V 

50.  Several  problems  relating  to  the  development  of 
solids  by  triangulation  are  given  on  this  plate.  The  same 
principles  that  were  shown  in  connection  with  the  develop- 
ment of  the  transition  piece  in  Figs.  28  to  31  are  used  in 
these  drawings;  but,  owing  to  the  different  shapes  assumed 
by  the  solid  selected  for  each  problem,  slightly  different 
constructions  are  given  in  each  case. 


54       DEVELOPMENT  OF  SURFACES      §  16 

Particular  attention  should  be  paid  to  the  manner  in 
which  the  triangles  are  located  on  the  irregular  surfaces, 
care  being  exercised  that  similar  points  in  each  view  shall  be 
taken  as  a  basis  for  finding  the  true  length  of  each  line. 


PROBLEM  16 

51.  To  develop  tlie  surface  of  an  Irregular  solid 
having  parallel  bases. 

The  solid  shown  in  perspective  in  Fig.  32  is  the  same  as 
that  used  in  Practical  Projection  for  the  projections  of  views 
of  an  irregularly  formed  solid  and  a  sectional  view  from  a 
given  cutting  plane. 

CoxsTRUCTiox. — The  first  step  is  to  draw  a  right  plan  and 
elevation,  as  shown  at  Fig.  1  {a)  on  the  plate.  Draw  the 
horizontal  diameter  a  m  through  the  plan,  thus  dividing  the 
solid  into  svmmetrical  halves.  It  will  now  be  seen  that,  if  a 
development  is  made  of  the  upper  portion  of  the  solid,  as 
seen  in  the  plan,  a  duplication  of  the  resulting  figure  will  be 
the  complete  development.  In  order  to  locate  the  sides  of 
the  triangles  that  are  to  be  assumed  on  the  surface  of  the 
solid,  the  outline  of  the  bases  is  divided  into  the  same  num- 
ber of  equal  parts  (in  this  case  6).  as  at  a,  c,  c,  etc.  on  the 
lower  base  and  d,  d,  f,  etc.  on  the  upper  base. 

Draw,  in  succession,  lines  alternately  from  the  points  on 
the  upper  and  lower  bases;    ab  being  on  the  line  of  the 

diameter,    draw    be,    c  d,    de,    etc. 

^1^        ^^  Project  these  points  and  lines  to  the 

jg/j     ~f^^^^^         elevation,    and    represent    them    by 
^i  /     /  1  ^^^^^'         dot-and-dash  lines,    as  used  in   pre- 
^■/.J-f— _t:  ceding    developments    to    designate 

^^''  _i^^fc^'    edge  lines.     The  surface  of  the  solid 

^"'^  is   thus    divided    into   a    number   of 

FIG.  32  triangles  that  may  be  better  under- 

stood by  the  student  from  an  inspection  of  Fig.  3-2.  This 
is  a  perspective  view  of  the  solid,  showing  in  the  full  lines 
the  triangles  that  have  been  assumed  on  its  surface.      The 


§  10      DEVELOPMENT  OF  SURFACES       55 

lengths  of  the  sides  b  d,  df,  ac,  rr,  etc.  may  be  taken  as 
chord  distances  directly  from  the  plan,  since  they  are  there 
shown  in  their  full  length.  A  construction  of  right-angled 
triangles  is  necessary,  however,  in  order  to  find  the  true 
lengths  of  the  lines  d  r,  cd,  de,  etc.,  Fig.  1  of  the  plate;  and 
in  order  to  construct  them,  the  base  line  a'  in'  of  the 'solid 
in  the  elevation,  Fig.,],  is  extended  indefinitely  towards  the 
right  of  the  drawing  to  ;//",  as  shown  at  (/;). 

At  a  convenient  distance  from  the  elevation  locate  a 
point  on  this  line,  as  at  //  in  Fig.  1  {b)  on  the  plate.  Take 
the  distance  be,  as  shown  in  the  plan,  and  set  it  off  with  the 
dividers,  as  at  b'  c' ;  in  like  manner,  make  c  d'  equal  to  cd, 
as  shown  on  the  plan,  and  proceed  to  copy  all  the  distances 
there  shown  until  the  point  m"  is  reached.  It  will  be  seen 
from  an  examination  of  the  projections  that  the  lines  a'  b 
and  ///'  n  are  shown  in  their  true  length  in  the  elevation, 
and  a  triangle  is,  therefore,  not  required  for  those  lines. 

Since    the    bases    of   the    solid  are   parallel,   the   vertical 
height  of  the  triangles  at   {b)   is  the  same  in  all  cases,  and 
may  be  projected  from  the  elevation  as  shown  on  the  plate. 
The  true  lengths  of  all  lines  now  being  determined,  the  tri- 
angles may  be  constructed  as  shown  at   Fig.  1   {c).     Draw 
the  line  ^/;,  making  it  equal  in  length  to  the  corresponding 
line  in  the  elevation— that  is,  a'  b;    next,    describe  an  arc 
from  rt-  as  a  center,  with  a  radius  ^r,  taken  from  the  plan  at 
{a) ;  intersect  this  arc  by  an  arc  described  from  ^  as  a  cen- 
ter, with  a  radius  equal  to  the  length  of  the  hypotenuse  of 
the  triangle  whose  base  is  b' c'  in   {b).     This   completes  the 
triangle  a  be  at  Fig.  1  (r).      The  triangle   bed  is  next  con- 
structed in  a  similar  manner,  and  the  completion  of  the  devel- 
opment is  accomplished  by  a  continuation  of  the  methods 
described.       Small  arrows  are  introduced  in   Fig.    1   (r)   to 
indicate  the   location  of  the   centers  of  the  corresponding 
arcs.    Thus,  the  arrowhead  on  the  line  be  is  pointed  towards  b, 
and  indicates  that  the  center  of  the  arc,  by  means  of  which 
the  point  c  is  determined,  is  located  at  b{ e  in  like  manner 
is    similarly  shown   to    be  the   center  of  the   arc  described 
through  {d).     Since  there  are  no  edges  to  be  bent  angularly 


56       DEVELOPMENT  OF  SURFACES      §  16 

when  the  surface  is  formed  to  the  shape  shown  in  the  plan 
and  elevation,  the  circular  indicators  are  omitted;  but 
arrowheads  indicating  the  boundary  edges  of  the  develop- 
ment may  be  added  as  heretofore,  completing  the  problem. 


PROBLEM  17 

52.  To  develop  tlie  surface  of  an  Irregular  solid 
liaving:  iuclmed  bases. 

A  perspective  view  of  this  solid  is  shown  in  Fig.  33 ;  it  is 
a  modification  of  the  solid  used  for  the  preceding  problem, 
and  may  be  drawn  as  shown  on  the 
plate  at  Fig.  2  {a).  The  outline  of 
the  lower  base  is  drawn  as  in  Fig.  1, 
and  the  center  of  the  semicircle  at  o 
is  also  the  center  of  the  circle  that 
represents  the  inclined  upper  base, 
the  angle  of  inclination  being  45°. 

F'^-  ^'^  Construction. — Draw  a  line  (not 

shown  on  the  plate)  vertically  upwards  from  o  and  fix  the 
point  // 1^  inches  above  the  lower  base  of  the  solid.  Through 
this  point  draw  the  line  d  n  at  the  given  angle ;  and  at  right 
angles  to  bn  describe  tbe  circle  representing  the  full  view 
of  the  upper  base,  as  shown.  Next,  locate  the  points  shown 
on  the  semicircle  and,  by  means  of  the  temporary  full  view 
shown  at  b'  h'  //',  project  the  inclined  view  of  the  upper  base 
in  the  plan.  These  points  are  then  used,  as  in  the  prece- 
ding problem,  for  the  purpose  of  defining  the  triangles. 
Divide  the  lower  base  of  the  solid,  as  shown  in  the  plan, 
into  an  equal  number  of  parts,  and  draw  lines  representing 
the  sides  of  the  triangles,  as  in  Fig.  1,  producing  them  in 
the  plan  and  elevation,  as  be,  c d,  dc,  etc.  Fig.  2  {a). 

The  manner  of  determining  the  true  lengths  of  the  lines  b  c, 
c d,  de,  etc.,  is  slightly  different  from  that  used  in  the  pre- 
ceding problem,  since  the  vertical  distances  are  not  the  same 
in  all  cases.  Produce  the  lower  base  a  m  to  the  left,  as 
shown  at  {b)  on  the  plate,  and  on  this  line  set  off  the  lengths 


§  1<;  DEVELOPMENT  OF  SURFACES  57 

of  the  lines  he,  c a\  d i\  etc.  as  they  appear  in  the  plan  at 
Fig.  -l  (a).  The  vertical  heights  are  then  projected  from 
the  elevation  in  the  manner  shcjwn,  taking-  similar  points  in 
each  case. 

The  true  lengths  of  all  lines  now  havin.g  been  determined, 
tlie  development  may  be  constructed  as  shown  at  Fig.  2  (c). 
In  this  case,  as  in  all  instances  where  the  solids  are  uneven 
and  irregular  in  their  form,  it  is  preferable  to  begin  the 
development  from  the  longest  edge  that  is  shown  in  its  true 
length.  The  line  /// //  is,  therefoi-e,  copied  as  shown  at  {c), 
and  the  triangle  //  jh  I  constructed  as  in  the  preceding 
development,  taking  the  lengths  of  the  radii  // /,  lj,j  h,  etc. 
from  the  full  view  of  the  upper  base,  and  the  lengths  of  the 
radii  ;//  /,  //',  kj\  etc.  from  their  respective  triangles  as 
formed  at  (/;),  while  in  k\  k  i,  i g-^  etc.  are  taken  from  the  full 
view  of  the  lower  base  as  shown  in  the  plan  at  {a). 


PROBLEM   18 

53.  To  develop  the  surface  of  an  Irregfiilar  solid 
\vhose  iippei'  base  is  vij>'litly  iucliued  and  Avliose  loAvex* 
base  Is  a  portion  of  a  cylinder. 

This  solid  is  shown  in  perspective  in  Fig.  34,  and  the  tri- 
angles that  are  to  be  located  by  the  student  in  the  projec- 
tions are  represented  in  the  drawings  shown  on  the  plate. 

Construction. — As  may  be  seen  from  Fig.  o  {a)  on  the 
plate,  the  projections  do  not  differ  materially  from  those  of 
the  preceding  problems.  First  draw 
the  oval  in  the  plan  as  in  the  two 
preceding  problems,  noting  that,  in 

this  case,  the  outline  is  a  foreshort-         ig.  Wliim^^ 

ened  view  of  the  real  surface.     Next,        '  ^' 
draw  a  line  vertically  upwards  from  o, 
the  center  of  the  semicircle,  to  the 
point   h  in   the   elevation,    which    is  fig.  ai 

3  inches  distant   from  o.     Through  //  draw  b  n  at  an   angle 
of  30°  with   the  horizontal ;  at   right   angles  to  b  n   project 
the  full   view  of  the   upper  base,  which,  as  in  Problem   17, 
M.  E.    v.— 22 


58       DEVELOPMENT  OF  SURFACES      §  16 

is  a  circle.  Through  the  plan  draw  a  in  and  bisect  it  at  x\ 
with  X  as  a  center  and  a  radius  of  l|-f  inches  describe  in 
the  elevation  the  arc  a  g  ni,  representing  that  view  of  the 
lower  base. 

As  in  the  preceding  problem,  project  the  foreshortened 
view  of  the  upper  base,  and,  as  before,  designate  the  posi- 
tions of  the  six  spaces.  Since  the  view  of  the  base  in  the 
plan  is  foreshortened,  it  is  necessary  to  project  its  full  view, 
in  order  to  ascertain  the  true  distance  around  the  outline. 
Divide  the  foreshortened  outline  of  the  lower  base  shown  in 
the  plan  into  six  equal  spaces,  at  a,  r,  e,  etc.  First,  how- 
ever, draw  the  center  line  a  in,  as  in  previous  cases,  and 
then  project  to  the  elevation  the  points  thus  located.  To 
produce  the  full  view,  as  at  Fig.  3  (r),  extend  the  line  a  ni 
in  the  plan  tOAvards  the  left  as  far  as  to  ;//',  and  on  this  line 
lay  ofif  the  stretchout  of  the  lower  base,  as  shown  in  the 
elevation ;  thus  that  portion  of  the  solid  is  treated  as  a  sur- 
face developed  by  means  of  parallel  lines.  Draw  edge  lines 
perpendicular  to  the  stretchout,  as  on  the  plate,  and  pro- 
duce developers  from  the  points  a,  r,  r,  g,  etc.  in  the  plan 
to  a\  c',e',  g' ,  etc.  at  (r).  The  light  curve  shown  at  {c)  is 
then  drawn  and  represents  the  true  outline  of  the  lower  base 
of  the  solid ;  measurements  may  now  be  taken  from  points 
on  this  outline,  as  a'  c\  c'  c\  etc.,  for  the  radii  of  the  arcs 
required  in  that  portion  of  the  development  at  (cz'),  their 
true  lengths  thus  being  shown.  The  true  lengths  of  the  lines 
shown  in  the  projections  at  be,  c  d,  d c,  etc.  are  obtained  as 
before  by  constructing  diagrams  of  triangles  at  (/;),  {b). 
Their  projection  on  both  sides  of  the  elevation  is  done  to 
avoid  confusion  from  having  a  number  of  lines  cross  on  the 
drawing. 

Note  the  varied  heights  of  the  triangles,  hence  the  need 
of  extreme  care;  for  if  corresponding  points  are  not  taken 
from  both  plan  and  elevation,  it  will  be  difficult  to  trace  the 
resulting  errors.  The  true  lengths  of  all  lines  having  been 
determined  in  (/?),  {b),  and  (r),  the  development  at  [d)  may 
be  constructed  by  methods  precisely  like  those  used  in 
Problems  10  and  17. 


§  IG      DEVELOPMENT  OF  SURFACES       59 

Since  iii  n  is  the  longer  true  edge  shown  in  the  elevation, 
the  development  should  begin  from  that  line  by  making  ;//  // 
at  (//)  equal  to  ;;/ ;/  in  the  elevation  at  {a).  AVith  a  radius  )i  I 
(taken  from  the  full  view  of  the  upper  base),  and  with  ;/  in 
{d)  as  a  center,  describe  an  arc  as  shown,  and  intersect  it 
at  /  with  an  arc  described  from  ui  as  a  center,  with  a  radius 
equal  to  the  hypotenuse  of  the  triangle  w/at  (/;)— the  true 
length  of  the  line  ////,  as  shown  at  {a). 

This  completes  the  triangle  m  ii  I  ixt  {d).  Next,  describe 
an  arc  from  ;;/  as  a  center,  with  a  radius  ;//'  k'  taken  from 
the  full  view  of  the  lower  base  at  (r),  and  intersect  this  arc 
at  /'  with  an  arc  described  from  /  as  a  center  and  a  radius 
equal  to  the  hypotenuse  of  the  triangle  Ik  at  (/;).  The  tri- 
angle ni  Ik  is  thus  completed,  and  the  remainder  of  the 
figure  at  {d)  is  constructed  in  a  similar  manner.  The 
development  at  {d)  is  one-half  of  the  irregular  surface  of 
the  solid  shown  in  Fig.  34. 

It  should  be  noted  that,  in  the  practical  work  of  laying 
out  patterns  by  this  method,  a  sufficient  number  of  points 
should  be  located  on  both  bases  of  the  solids  to  insure  accu- 
racy in  tracing  the  curved  line  of  the  development. 

54.     Impoi'tanoe    of   a    Correct    Projection. — It    will 

be  seen  from  the  foregoing  problems  that  in  each  case  the 
first  operation  is  the  drawing  of  a  correct  projection.  The 
true  lengths  of  all  real  or  assumed  lines  that  may  have  been 
shown  in  such  a  drawing  of  the  solid  are  thus  ascertained, 
and  the  draftsman  is  then  enabled  to  determine  which  of 
the  three  general  methods  is  to  be  applied  in  order  that  a 
development  may  be  produced.  The  views  to  be  drawn  are 
in  all  cases  those  that  will  represent  the  solid  in  a  right  posi- 
tion, since  the  true  length  of  any  line  is  most  easily  obtained 
from  such  drawings.  The  ease  with  which  this  is  accom- 
plished is  clearly  shown  in  the  foregoing  examples;  and 
if  the  student  will  devote  his  attention  to  the  projection 
of  a  variety  of  commonly  occurring  trade  subjects,  he 
will  quickly  acquire  a  facility  obtained  only  by  constant 
practice. 


60 


DEVELOPMENT  OF  SURFACES 


§16 


Particular  attention  is  directed  to  the  imaginative  feature 
of  the  projection,  as  mentioned  in  Practical  Projection ;  for 
a  correct  conception  of  the  actual  position  of  the  various 
lines  as  they  will  appear  in  the  completed  object  can  be 
acquired  in  no  other  way,  and  it  is  of  extreme  importance 
to  the  draftsman  desiring  to  become  proficient  in  the  pro- 
duction of  developments.  If  a  student  is  expert  at ' '  reading  " 
drawings,  he  will  experience  no  difficulty  in  applying  the 
various  principles  that  have  been  given  for  producing 
developments  of  surfaces. 

C>^,  Employment  of  Modified  Metliods. — In  produ- 
cing the  developments  given  in  the  last  three  problems, 
the  triangles  are  not  always  projected  from  the  views  in  the 
manner  shown  on  the  plate.  Short  methods  are  often  used  ; 
but  since  in  such  cases  there  is  more  chance  for  error,  the 
student  will  do  well  not  to  attempt  any  other  method  than 


Fig.  35 

that  just  described — at  least,  not  until  the  principles  are 
firmly  fixed  in  his  mind.  As  the  drawings  from  which  such 
developments  are  made  are  often  too  large  to  permit  the  pro- 
jection of  the  triangles  in  the  manner  shown,  the  construc- 
tion given  in  Fig.  35  is  often  applied.  In  this  figure  the 
triangles  for  Problem  16  are  shown  at  (^),  and  those  for 
Problems  17  and  18,  at  {b)  and  (<:),  respectively.     A  single 


§  IG      DEVELOPMENT  OF  SURFACES       61 

right  angle  is  first  drawn,  as  a  be;  the  lengths  of  all  lines 
shown  in  the  plan  are  then  set  off  with  the  dividers  on  the 
side  b  c,  while  the  vertical  distances  taken  from  the  elevation 
are  set  off  on  the  side  a  b.  A  number  of  slanting  lines  are 
then  drawn  between  the  points  thus  located,  each  forming 
the  hypotenuse  of  its  respective  right-angled  triangle. 
Unusual  care  must  be  observed  when  this  method  is  adopted, 
since  the  points  are  located  close  together  and  are  very  apt 
to  be  mistaken  for  one  another. 

When  many  lines  thus  appear  in  the  drawing,  the  student 
is  very  apt  to  become  confused  and  to  consider  the  drawing 
as  complicated.  If  due  care  is  used  and  ample  time  taken, 
this  confusion  will  be  avoided,  for  the  drawings  themselves 
depend  on  the  simplest  of  principles — principles  that  may 
readily  be  understood  by  any  one  willing  to  give  the  time 
necessary  for  mastering  this  work. 

.K).  Scalene  Cone. — The  development  of  this  solid 
illustrates  a  short  method  of  triangulation  applicable  to  a 
number  of  solids,  particularly  to  those  represented  by  transi- 
tion pieces  one  of  whose  bases  is  a  polygon.  A  right  view 
of  such  a  cone  is  presented  in  Fig.  36  (a),  and  an  inspection 
of  its  elements  will  show  that  they  are  of  unequal  length. 
This  inequality,  combined  Avith  the  fact  that  a  section  taken 
at  right  angles  to  the  axis  of  a  scalene  cone  is  not  a  circle, 
precludes  its  development  by  the  method  applied  to  radial 
solids,  although  the  process  is  a  combination  of  that  method 
and  triangulation.  In  accordance  with  instructions  given 
hereafter,  a  drawing  of  this  development  should  be  made  by 
the  student,  although  it  is  not  required  to  be  sent  to  the 
Schools  for  correction. 

After  drawing  the  right  plan  and  elevation  as  in  Fig.  36  {a), 
the  base  is  divided  into  a  convenient  number  of  equal 
parts  (1"^  in  Fig.  36).  Next,  the  elements  are  drawn  in  both 
views,  and  it  may  then  be  seen  that  the  surface  of  the  cone  is 
divided  into  a  number  of  triangles  whose  common  apex  is 
at  o,  the  vertex  of  the  cone.  In  the  elevation,  the  elements  <;_^ 
and  oa  are  the  only  ones  shown   in  their  true  length.     In 


62 


DEVELOPMENT  OF  SURFACES 


16 


order    to    determine    the    true    lengths    of    the    remaining 
elements,  the  drawing  shown  at  Fig.  36  {d)  is  constructed. 


\\  \  \  ■  X^  V     /      /  / 


Fig. 36 


Draw  the  right  angle  /  ;//  //  and  set  off  ;//  o,  the  vertical 
height  of  the  cone. 

The  length  of  each  element  shown  in  the  plan  at  (a)  is 
now  set  off  on  the  line  ;//;/;  thus,  make  //i ^  at  (/?)  equal 
to  ^'^  in  the  plan  at  (a),  inf  at  (/-')  equal  to  o'  f  zX.  {a),  etc. 
DraAv  0 g,  of,  o  e,  etc.  at  {b),  thus  producing  the  elements  of 
the  scalene  cone  in  their  true  length,  the  method  used,  as 
Avill  be  seen  by  the  student,  being  similar  to  that  used  in  the 
other  triangulation  problems. 

From  o  as  a  center,  with  o g,  of,  o  c,  etc.  as  the  respective 
radii,  describe  arcs  in  the  manner  shown  in  Fig.  36  {b).  At 
a  convenient  point  locate^',  and  draw  g'  o  as  indicated  at  (r). 
This  line  {g  o)  is  one  edge  of  the  development,  which  may 
be  completed  by  making^'-'/'  at  (f)  equal  togfm  the  plan 
at  {a),f'  c'  at  (r)  equal  to  f  c  at  {a),  etc.,  the  compasses 
being  set  at  this  distance  and  similar  arcs  described  as  the 


§  IG      DEVELOPMENT  OF  SURFACES       63 

successive  points  are  located.  Proceed  in  the  manner  shown 
until  the  point  ^'■"  is  reached,  when  the  development  is  com- 
pleted by  drawing  o g"  and  tracing  the  curve  through  points 
at  the  intersection  of  the  arcs.  The  outer  edge  lines  of  the 
development  may  be  further  noted  by  the  use  of  indicators 
as  before  described. 

There  are  many  irregular  solids  to  which  this  method  of 
development  may  be  applied.  The  student  should  therefore 
work  out  this  drawing  very  carefully,  that  the  construction 
may  be  well  understood  and  the  method  of  its  application 
readily  seen.  Where  convenient,  models  should  be  made  of 
these  developments.  These  models  may  be  cut  out  of  paper 
or  thin  sheet  metal  and  afterwards  formed  up  to  represent 
the  solids  described  in  the  particular  projections  accompany- 
ing each  problem.  xV  better  idea  of  the  solids  and  their 
respective  developments  will  thus  be  obtained.  Should  any 
errors  arise  in  the  course  of  the  work,  they  may  frequently 
be   detected   by   this   means. 


approxi:mate  dea^i^opmexts 

57.  The  Spliere. — The  sphere  is  the  most  prominent 
example  of  the  many  solids  whose  surfaces  admit  of  approxi- 
mate development.  The  methods  used  with  solids  capable  of 
accurate  development  must  be  applied  to  solids  of  approxi- 
mate development.  Since  solids  of  approximate  develop- 
ment may  be  resolved  into  parts  resembling  those  capable 
of  accurate  development,  it  is  clear  that  the  same  general 
methods  are  easily  applicable.  Thus,  in  the  case  of  the 
sphere,  that  solid  is  resolved  into  a  number  of  frustums  of 
cones,  in  the  manner  described  in  the  next  article. 

As  already  stated,  it  is  necessary  to  submit  the  covering 
for  these  solids  to  the  operations  of  "  raising,"  in  order  to 
conform  the  metal  to  the  profile  of  the  solid  itself;  an  allow- 
ance is  therefore  required  for  the  consequent  stretching  of 
the  stock.  This  allowance  varies  with  the  thickness  and 
quality  of  the  material ;  hence,  no  general  rules  can  be  given 


64       DEVELOPMENT  OF  SURFACES      §  16 

here.     The  mechanic  must   be  acquainted  with  the   nature 

of  the  material  in  order 
to  determine  the  allow- 
ance required  in  each 
case. 

58.        Developraent 
of  tlie  Sphere. — It  has 

been  stated  that  each 
solid  of  approximate 
development  must  be 
referred  to  one  of  the 
three  classes  of  solids 
capable  of  accurate  de- 
velopment. It  is  first 
necessary,  therefore,  to 
determine  which  of  the 
regular  solids  the  irreg- 
ular solid  to  be  devel- 
oped most  resembles. 
The  blanks,  or  pat- 
terns, produced  by 
these  methods  are,  as 
has  been  observed,  fiat, 
or  plane,  surfaces;  and 
since  it  is  necessary  to 
''raise"  these  surfaces 
by  hammering,  the 
draftsman  must  imag- 
FiG.  37  ine  the  surface  desired 

in  the  final  form.  The  principle  by  which  these  solids  are 
classified  is  illustrated  in  Fig.  37  by  the  perspective  of  the 
sphere  and  its  resolved  cones.  After  the  sphere  has  thus 
been  resolved  into  cones — or  rather,  frustums  of  cones — 
their  developments  may  be  produced  in  the  regular  way, 
each  frustum  being  separately  treated,  in  the  manner  shown. 
This  is  called  development  bj  ::oHes,  and  may  be  applied  to  a 
number  of  irregular  solids  resembling  the  sphere. 


16 


DEVELOPMENT  OF  SURFACES 


05 


Another  method  of  treatment  is  shown  in  Fig.  38,  certain 
sections  of  the  sphere  being"  here  considered  as  portions  of 

the  surface  of  the  cylinder.  This 
method  is  referred  to  as  the  develop- 
ment by  gores ;  the  patterns  are 
developed,  as  with  any  portion  of 
the  cylinder,  by  means  of  parallel 
lines. 


Development  of  Gore  A 

When  the  curved  surface  of  any 
solid  is  such  that  it  cannot  be  con- 
sidered as  part  either  of  a  cone  or 
of  a  solid  with  parallel  lines,  it  is 
customary  to  determine  a  series  of 
sections  in  a  regularly  occurring  order  and  apply  the 
method  of  triangulation  as  shown  in  Problems  16  to  18. 
Such  surfaces  are  then  formed  up  as  nearly  as  possible  to  the 
shape  of  the  solid  and  are  then  submitted  to  the  operations 
referred  to  for  "  raising  "  to  their  proper  shape. 


Fig.  38 


INDEX 


Note  —All  items  in  this  index  r 
section.  Thus,  "  Addendum  14  51' 
of  section  14. 

A  Sec. 
Abbreviations  used  on   draw- 
ings    1-t 

Accuracy,  Importance  of 15 

"  Importance  of 16 

Accurate  developments 10 

Addendum 1-1 

Alphabet,  Block-letter 13 

Angle  of  projection    15 

To  bisect  an   13 

with    a    given    line,   To 

draw 13 

Approximate  developments....  10 
"              developments —  16 
Arc  method  of  secondary  pro- 
jectors   15 

"    of  circle  equal  in  length  to 

given  straight  line 13 

"    To  find  center  of  an 13 

Assembly  drawings 14 

Axial  line IS 

A.\is  of  helix 13 

of  parabola  13 

B  Sfc. 

Backlash 1^ 

Base  circle 1-1 

"     line 15 

Basis  of  projection 15 

Bent,  Definition 14 

Bevel  gear-wheels 14 

"      gears.     Developing    the 

teeth  of  14 

"      gears.  How  to  draw 14 

Bisect  an  angle,  To 13 

"      a  straight  line,  To 13 

Block-letter  alphabet 13 


efer  first  to  the  section  and  then  to  the  page  of  the 
'  means  that  addendum  will  be  found  on  page  51 


Pu£-e 


41 

38 
11 
63 
46 
45 

51 
54 
10 
41 


Sec.  Page 


Blueprinting 14 

"  frames 14 

Board,  Drawing 13 

Bolts,  Square-headed    13 

Bore,  IMeaning  of 14 

Bow-pen 13 


pencil 


13 


Box,  Clamp 14 

Brace-blocks —  14 

Brake 14 

"      14 

"      lever 14 

Brakes 14 

Brass  nipple 14 


Calculations,  Definitions  and. . 

Carrier 

Ceiling  plan 

Center  line 

"        line 

"        lines 

"  lines.  Importance  of..  . 
Chord   distance,   Measurement 

of 

Circle 

"      Base 

"      Generating 

"      Involute  of  the  

Circles,  Methods  of  shading.... 

Pitch .   . 

"        Route 

Circular  pitch 

Circumference,     To     pass, 

through  three  points  not  on 

the  same  straight  line 


Sec. 

Pag~ 

14 

55 

14 

24 

15 

9 

13 

53 

15 

59 

14 

1 

14 

~ 

10 

34 

15 

81 

14 

54 

14 

48 

14 

50 

13 

76 

14 

50 

14 

57 

14 

50 

38 


Vll 


INDEX 


Sic.  Page 

Clamp U  24 

"      box 14  21 

Classification  of  solids        IC  3 

Cloth,  Tracing H  77 

Compasses 13  5 

CompKmnd  trestle 14  27 

Combinations  of  section  lines..    14  3 
Cone  and    cylinder,    Intersec- 
tion of 15  95 

'•      and    cylinder.    Intersec- 
tion of 13  CC 

'*      and  pyramid.  Relation  of    16  36 

"       cut  by  plane 13  65 

"       Envelopment  of 16  44 

"      Development  of  frustum 

of 16  46 

*'      Development    of    inter- 
secting     16  47 

"       Development  of  irregu- 
lar frustum  of 16  46 

"      Development  of  scalene .     16  61 

'■       Elements  of 13  65 

'•       Elements  of 15  82 

"      Intersection  of 15  96 

"       Projection  of 15  68 

Conic  sections 13  65 

"      sections 15  81 

"      sections.  Drawing  Plate.    13  C5 
Conical  surface,  Developmentof  13  ~2 
Construction  lines  for  develop- 
ments     16  18 

Conventional  representation  of 

a  nut  14  15 

representation  of 

screws 14  10 

Cored,  Meaning  of 14  10 

Coupling,  Flange 14  24 

Shaft  flange 14  23 

Crank 14  21 

Crated.  Meaning  of 14  10 

Cube,  Development  of 16  15 

Projection  of 15  55 

"      Sections  of 15  71 

Curves.  Irregular 13  15 

Cutting    plane.     How     repre- 
sented      15  72 

Cycloid 14  48 

Cycloidal  teeth 14  51 

Cylinder 13  56 

13  64 

"         and  cone.  Intersection 

of ..     13  66 

"         and  cone.  Intersection 

of 15  » 

"         and  prism.    Intersec- 
tion of .  .    15  91 


Sec.  Page 
Cylinder  and  prism.  Intersec- 
tion of 16  35 

and  sphere.  Intersec- 
tion of 15  99 

and  sphere.  Intersec- 
tion of 16  35 

Development  of 16  27 

Developmentof  iuter- 

secting 16  35 

Intersecting 15  93 

Intersection  of 13  66 

Intersection  of ...  10  35 

'■         Projection  of 15  94 

"         Sections  of 15  65 

Cylindrical  ring.  Cast-iron 13  59 

"           surfaces.  Develop- 
ment o: 13  71 

surfaces.  Intersec- 
tion of  two 13  68 

D  Sec.  Page 

Dark  surface 13  74 

Definitions  and  calculations. . .  14  55 

Detail  drawings 14  11 

Developers IG  18 

Developing  the  teeth  of  bevel 

gears 14  61 

Development 16  1 

.\ccurate 16  1 

"              Approximate 16  5 

Approximate 16  63 

by  parallel  lines.  16  5 

"             by  parallel  lines.  16  13 

"              by  radial  lines. ..  16  8 

"             by  radial  lines...  16  36 

'•             by  triangulation  16  9 

by  triangulation  16  49 

'•             Finishing  of 16  19 

"  Importance       of 

certain  views  in  16  !3 

'*             Method  used  in..  16  2 

"              of  cone 16  44 

"             of  conic  frustum  16  46 
"             of    conical    sur- 
faces   13  72 

'■             of     conical     sur- 
faces   16  44 

"             of  cube 16  15 

"             of  cylinder 16  27 

"  of  cylindrical 

surfaces 13  71 

of  cy  li  n  drical 

surfaces 16  27 

•'             of     frustum      of 
^   hexagonal  pyr- 
amid   16  46 


INDEX 


IX 


Sec. 
Development  of   intersected 

cylinder Ifi 

"  of    intersected 

cylinder IG 

of    intersected 

prism 16 

"  of    intersected 

prism IG 

"  of    intersected 

solids 16 

"  of      intersecting 

cones IG 

"  of      intersecting 

cylinder     and 

sphere 16 

"  of      intersecting 

cylinders 16 

"  of      intersecting 

prisms 16 

"             of  intricate  solids    16 
"             of    irregular    in- 
tersecting   cyl- 
inders      16 

"             oi' irregular  solid    16 
"              of  irregular  solid     16 
"             of  irregular  solid    16 
"             of  octagonal  pyr- 
amid      16 

"  of  pentagonal 

prism 16 

"  of  pyramid 16 

"  of  pyramid 16 

"  of  pyramid 16 

"  of  pyramid 16 

"  of  quadrangular 

pyramid 16 

"  of  scalene  cone. .     16 

"  of  sphere 16 

"             Position    of  par- 
allel        16 

"  Position  of  radial     16 

"  Rules  for  parallel     16 

"  Rules  for  radial.     16 

Developments— I,    Drawing 

Plate 16 

'■  II,       Drawing 

Plate 16 

"  III,     Drawing 

Plate  16 

"  IV,     Drawing 

Plate 16 

"  V,       Drawing 

Plate 16 

"  Intersections 
and  ;  Draw- 
ing Plate 13 


Pa^e 

28 
31 
28 
20 
47 

35 

29 

31 
32 


Sec. 

Diameter  at  root 14 

"  over  all 14 

Different  views,  To  draw l.i 

Dimension  lines 15 

Dimensions 13 

Distance  piece 13 

Dividers 13 

Division  of  solids.  How  accom- 
plished      16 

Dog   11 

Double  square-threaded  screws    l4 

"        V-threaded  screw 14 

Drawing  board   13 

"         Definition  of 13 

Detail 14 

Freehand 13 

ink 13 

"         Instruments  and  ma- 
terial required  for. .     13 
"         Reading  a  working..     14 

"         Mechanical  13 

paper 13 

"         pen,  To -sharpen  a 13 

"         pencils 13 

"         Perspective 15 

"         Plate,  Developments 

-I 16 

"         Plate,  Developments 

-II 16 

"         Plate,  Developments 

-III 16 

"         Plate,  Developments 

-IV 16 

"         Plate,  Developments 

-V 16 

"         Plate,  Intersections — 

I 15 

"         Plate,  Intersections — 

II 15 

"  Plate,  Projections— I.  15 
"  Plate,  Projections— II  15 
"         Plate,    Projections  — 

III 15 

"  Plate,  Sections— I 15 

"  Plate,  Sections— II...  15 
"         Plate,   Title  :    Bench 

Vise 14 

"         Plate,     Title:     Bevel 

Gears 14 

Plate,    Title:     Brush 

Holder 14 

Plate,  Title  :  Commu- 
tator      14 

Plate,  Title:  Details.     14 
Plate,   Title  :    Eccen- 
tric andBrakeLeve.    14 


Pas;e 
50 
50 
15 
14 
60 
58 
9 

10 
24 

17 
18 

1 

1 
11 

1 
13 

1 

66 

1 

10 
14 
10 
3 


INDEX 


Sec. 
Drawing  Plate,  Title  :    Flange 

Coupling 14 

Plate,  Title  :  Machine 

Details 14 

Plate,    Title :     Com- 
pound Rest 14 

Plate.  Title  :  Profiles 

of  Gear-teeth 14 

Plate,     Title:    Shaft 

Hanger  14 

Plate,      Title  :      Six- 
horsepower     Hori- 
zontal   Steam    En- 
gine     ...  14 

Plate,    Title  :      Spur 

Gear-wheels 14 

"  Plate,  Title :  Steel 
Columns  and  Con- 
nections    14 

Plate,  Title  :  Timber 

Trestle 14 

Plate,  Title  :   Turret- 
Lathe  Tools 14 

"         Projection 13 

"         To  read  a 15 

"         Use  of  a  section 15 

"         Working 15 

Drawings,  Abbreviations  used 

on 14 

"  Assembly 14 

"  General 14 

"  Kinds  of  working. .  14 

Drill,  Meaning  of 14 

Driving  fit 14 

E  S^c. 

Edge  lines —  16 

"      lines 16 

"      lines,  True 16 

Egg-shaped  oval  1.3 

Elbow,  Pattern  for  two-pieced  16 

Elements  of  cone 13 

"         of  cone 15 

Elevation 13 

15 

Front 13 

Side 13 

Ellipse 15 

"      13 

"      Methods  of  drawing  an.  13 

Epicycloid 14 

Equilateral  triangle.  To  draw.  13 

F  Sec. 

Face  ...    14 

"    Meaning  of 14 

Field  rivets 14 


Pa.?-e 
24 
21 
75 
48 
45 

84 
56 


11 
11 
10 
10 
10 

Page 
17 
25 
41 
42 
25 
65 
82 
52 
9 
52 
52 
82 
43 
43 
49 
36 

Page 
51 
.10 
35 


Sec.  Page 

Fillister  head   14  38 

Finishing  a  development 16  19 

First- angle  projection 15  7 

"      angle   projection,    Read- 
ing   15  56 

Fit,  Driving 14  10 

"    Forcing 14  10 

"    Shrinking 14  10 

Flank 14  51 

Flange  coupling 14  24 

"        coupling.  Shaft 14  23 

Forcing  fit 14  10 

Foreshortened  views 15  10 

Forms,  Geometrical 15  11 

"        Recognition  of  radial..  16  49 

Frames,  Blueprinting  ..  14  80 

Freehand  drawing 13  1 

Front  elevation 13  52 

"      elevation 15  9 

Frustum,   Development  of 

conic 16  46 

"           of    cone.     Develop- 
ment of  irregular..  16  46 

"  of  pyramid.. 13  49 

of   pyramid.   Devel- 
opment of ...  16  46 

Full  view  15  43 

"    view.  Position  of 15  52 

G  Sec.  Page 

Gear-wheels,  Bevel 14  55 

"     wheels.  Miter 14  55 

"     wheels.  Spur 14  55 

General  drawings 14  11 

Generating  circle 14  48 

Geometrical  forms 15  11 

Girts 14  31 

Gland  14  23 

Gores,  Pattern  for  sphere  by..  16  65 

H  Sec.  Page 

Hand  wheel 14  20 

Head  fillister 14  38 

Helix,  Pitch  of 13  46 

"      To  draw  a 13  46 

Hexagon,    To    inscribe    in    a 

circle 13  39 

Hexagonal  prism 13  56 

"  prism 13  63 

"  pyramid 13  57 

."  pyramid.  Develop- 

ment of   frustum 

of 16  46 

Hidden  screw  threads 14  8 

surfaces 15  58 

Holder,  Hollow  mill   14  37 

Releasing  die 14  37 


INDEX 


XI 


Sec.  Page 

Hollow  mill  holder 14       37 

Hoi  izontal  plane   15        17 

Hyperbola 13        66 

Hypocycloid 14       50 


I 


Ima]u;ination,  Help  of  the 

Importance  of  accuracy 

'■            of  correct  projec- 
tion  

Inclined  lines  

"         planes 

"         solids 

Ink,  Drawing 

Inkinjj;  drawings 

Instiumental  drawing 

Instruments  

Interedge  lines 

"  lines 

Intersection  of  cylinder 

"  of     cylinder     and 

cone 

"  of    cj'linder    and 

cone 

"  of    cylinder     and 

prism 

"  of     cylinder     and 

sphere 

"  of  prism 

"  of  solids 

"  of  two  cylindrical 

surfaces 

Intersections — I,  Drawing  Plate 
II,     Drawing 

Plate  

"  and  D  e  V  e  1  o  p  - 
ments,  Draw- 
ing Plate 

"  General  instruc- 
tions for 

'■  of  cones 

"  of  cylinder 

Intricate  solids.   Development 


Sec. 

Page 

15 

13 

15 

46 

10 

59 

15 

26 

15 

40 

15 

61 

13 

13 

13 

11 

13 

1 

13 

1 

16 

a5 

16 

23 

13 

64 

13 

66 

15 

95 

15 

91 

15 

99 

15 

88 

16 

20 

13 

68 

15 

87 

ot. 


Involute 

"         of  a  circle  

Irregular  curves 

'■  intersection  of  cylin- 
ders, Pattern  for... 

"  solid,  Development 
of 

"  solid,  Development 
of 

"  solid,  Development 
of 

"  solid,  Projection  of.. 


15 

92 

15 

96 

15 

93 

16 

32 

14 

50 

14 

50 

13 

15 

16 

3.3 

16 

54 

16 

56 

16 

57 

15 

78 

Irregular  solid,  Section  of 

Irregularly    outlined    surface. 

Projection  of. . . . 
"  outlined    surface, 

Projection  of. . .. 


Sec. 
15 


Page 


48 


15       54 


J  Sec.  Page 

Joint  riveted 14       23 

K  .5"^^.  Page 

Kinds  of  working  drawings 14        10 

L  Sec.  Page 

Laying  ofif  the  stretchout 16        16 

Letter,  Block 13       28 

Lettering 13        18 

Lever,  Brake 14       27 

Light  surface 13       74 

Line,  Axial 15       63 

"      Base 15        16 

"      Center 15       59 

"      Center 13       53 

"      Development  by  radial. .     16  8 

"      Development  by  radial. .     16        36 

"      Dimension 15       14 

"      Edge 16        17 

"      Edge  16  25 

"      in  a  development.  Finish 

of 16       19 

"      Inclined  position  of 15  26 

"      Miter 15  87 

"      Oblique  position  of 15  33 

"      of  sight 15  5 

"      of  sight.  Foot  of 15  17 

"      Projection  of 15  21 

"      Right  position  of 15  21 

"      To  bisect  a  straight 13  28 

"      To  divide  into  equal  parts    13  34 

"      To  draw  a  parallel  to  ...     13  30 
"    'To  draw  a  perpendicular 

to 13  29 

"      To  draw  a  perpendicular 

to 13  30 

"      To   find   the  true  length 

of 15  35 

"      True  edge 16  41 

Lines,  Center 13  53 

"      Center 14         1 

"      Center 15  59 

"      Character  of 13  48 

"      Interedge 16  23 

"      Interedge 16  25 

"      Pitched 14  59 

"      Shade 13  73 

"      used  in  drawing 13  48 

Lining,  Sections  and  section...     14         2 


Xll 


INDEX 


M  Sec.  Pa.s^e 
Measurement    of    chord    dis- 
tances   10  34 

Meclianical  drawing 13  1 

Method  of  triangulation 16  49 

Miter  line 15  8? 

"     gear-wheels 14  65 

Modified  methods  of  triangula- 
tion   16  GO 

N  Sec.  Pag-e 

Nipple,  Brass 14  18 

Nut,  Conventional  representa- 
tion of  a 14  15 

O  Sec.  Page 

Objects,  Repeated  parts  of 14  8 

"         Representation  of    ...  13  48 

Oblique  irregular  surface •    15  54 

position  of  line 15  33 

"        position  of  plane 15  49 

"        position  of  solid 15  61 

Observer,  Position  of 15  4 

Octagon  in  a  circle.  To  inscribe 

an   13  40 

Octagonal  prism.  Intersected. .  10  28 
"          pyramid.     Develop- 
ment of 16  45 

Ordinate  of  parabola 13  45 

Orthographic  projection 15  1 

Oval,  Egg-shaped 13  4d 

P  Sec.  Page 

Paper,  Drawing 13  10 

Tracing 14  77 

Parabola 13  66 

15  84 

To  draw  a 13  45 

Parallel     development.     Rules 

for 16  24 

"          lines,     Solids    devel- 
oped on 16  5 

"  to  a  straight  line.  To 

draw  a 13  30 

Parallelogram,  To  draw  a 13  37 

Pattern  for  T  joint  ..   16  25 

"       for  two-pieced  elbow..  16  25 
"        Relation    of     surfaces 

in  a  16  1 

Pen,  Ruling  or  right-line 13  11 

"    To  sharpen 13  14 

Pencils,  How  .sharpened 13  10 

"        used  in  drawing 13  10 

Pentagon  in  a  circle.    To    in- 
scribe a 13  39 

Pentagonal     prism.     Develop- 
ment of IG  26 


Sec. 
Perpendicular  to  a  straight  line, 

To  draw  a 13 

'■  to  a  straight  line, 

To  draw  a.  ...  13 

Perspective  drawing 15 

Pinion 14 

Pitch 14 

"      circles 11 

"      Circular 14 

"      of  helix  13 

"      lines 14 

Plan  13 

15 

Plane  15 

"      Cone  cut  by 13 

"      Cutting 15 

"     cutting  cylinder 15 

"      cutting  irregular  solid. . .  15 

"      cutting  prism   15 

"      cutting  pyramid 15 

"      Inclined,  Position  of 15 

"     Oblique,  Position  of 15 

"     of  projection 15 

"     Projection  of 15 

"      surfaces 15 

"      To  determine  position  of  15 

Planed,  Meaning  of 14 

Plate 13 

"      1 13 

"      I,  Hints  for  13 

"      II 13 

"      III 13 

"      IV 13 

"      V 13 

"       Developments — 1 16 

"      Developments— II 16 

"      Developinents — III 16 

"       Developments — IV 16 

"       Developments — V 16 

"      Intersections — I 15 

"      Intersections — II 15 

"      Projections— 1 15 

"      Projections — II 15 

"      Projections— III 15 

"      Sections — I ^  15 

"      Sections — II 15 

Plates,  Preliminary  directions 

for  drawing 13 

Point,  Projection  of 15 

Polygon    in   a    circle.    To    in- 
scribe a 13 

"  To  construct  a 13 

Position  of  full  views 15 

"        of  observer 15 

"         of     parallel     develop- 
ments     16 


Page 

29 

30 
3 

59 
60 
50 
50 
46 
59 
51 
t 
36 
65 
73 


76 
40 
49 
IS 
16 
36 
37 
10 
26 
27 
32 
33 
37 
39 
42 
25 
31 
33 
44 
53 
87 
93 
15 
47 
63 
70 
81 

26 
20 

40 
41 
52 

4 


INDEX 


XUl 


Sec.  Piige 
Position  of  plane,   How  deter- 
mined       15       37 

"        of    radial   develop- 
ments     16       A\ 

"        of  sections 15         9 

Preliminary      directions     for 

drawing  plates 13        26 

Primary  projectors 15        18 

Prism  and  cylinder.  Intersec- 
tion of 15        91 

"       and  cylinder,  Intersec- 
tion of 1(3        35 

"       Hexagonal 13        50 

Hexagonal 13        63 

"      Intersected 16  35 

"       Intersecting 15  88 

"       Intersecting 16  31 

"       Pentagonal 16  26 

"       Projection  of 15  59 

"      Rectangular 13  62 

"      Sections  of 15  75 

Profile  of  a  tooth 14  53 

Projection 13  50 

"  Angles  of )5  7 

"  Basis  of 15  41 

"  drawing    13  49 

"            drawing,  Proof  of. .     15  25 
"           Importance  of  cor- 
rect      16  59 

"  of  cone 15  08 

"  of  cube 15  55 

"  of  cylinder 15  94 

"  of  intersecting 

prisms 15  88 

"  of  irregular  solids.     15  78 

of  line 15  21 

of  plane 15  38 

of  point 15  20 

"  of  prism 15  59 

"  of  pyramid 15  76 

"  of  scalene  cone.  .   .     15  85 

"  of  solids    15  55 

"  Orthographic 15         1 

"  Views  necessary  in    15         6 

Projections,  Planes  of  15  16 

"  — I,  Drawing  Plate    15  15 

"  — II,      Drawing 

Plate 15  47 

"  — III,    Drawing 

Plate 15  &3 

Projectors 15  18 

"           Arc  method  of  sec- 
ondary       35  42 

"  Primary 15  18 

"  Secondary  15  18 

Proof  of  projection  drawing...     15  25 
M.   E.     v.— 23 


Sec.  Pag-e 

Proportioning  the  scale        14  14 

Protractor 13  17 

Pyramid  and  cone.  Relation  of  16  36 

"  Development  of  16  44 

"  Development  of 16  45 

"■  Development  of  ....  16  46 

"  Development  of 16  57 

"          Development  of  frus- 
tum of 16  46 

"  Frustum  of 13  49 

"■  Hexagonal 13  57 

"  Hexagonal 13  63 

"  Projection  of 15  76 

"  Sections  of 15  75 

0  Sec.  Paj^e 
Quadran.gular  pyramid.  Devel- 
opment of 16 


R 

Radial  developments 

"       forms     

"       lines.  Development  by. 

"       solids.  Revolution  of.. .. 

"      solids,  Stretchouts  for.. 

Rear  elevation 

Rectangular  prism 

Relation  of  pyramid  and  cone. 
Revolution  of  radial  solids  .... 
Right-line  pen   ...         

"      position  of  line   

Ring,  Cast-iron  cylindrical  ... 

Rivet 

Roof  plan 

Rules  for  parallel  development 

"      for  radial  development.. 

Ruling  pens 


44 

Sec.  Pug-e 
16         8 


16 
10 
16 
16 
15 
13 
16 
16 
13 
15 
13 
13 
15 
16 
16 
13 


S  Sec 

Scale 13 

'■      Proportioning  the         11 

Scales 14 

"      Kinds  of        14 

Scalene  cone.  Development  of.  16 

"        cone.  Projection  of  .. .  15 

Screw,  Double  square-threaded  14 

"       Single  V-threaded.. .    .  14 

.     "       Double  V-threaded.    ...  14 

*'      threads.  Hidden 14 

Screws,     Conventional     repre- 
sentation of 14 

Secondary  projectors 15 

"  p  r  o  j  e  c  t  o  r  s ,  A  re 

method  of  15 

Section  drawings 15 

"        drawings.  Use  of 15 


49 
30 
40 
39 

9 
62 
36 
40 
11 
21 
59 
57 

9 

ai 

42 
11 

Paf^e 
17 

14 
12 
12 
61 
85 
17 
17 
18 
8 

16 
18 

42 

9 
70 


XIV 


INDEX 


Sec. 

Section,  How  represented 15 

"         lines,  Combinations  of  14 

"         lining,  Sections  and...  14 

"        of  cube 15 

"         of  cylinder 15 

"         of  irregular  solid 15 

"         of  prism 15 

"         of  pyramid 15 

"         of  scalene  cone 15 

"         of  sphere 15 

"         Position  of 15 

Sections,  Conic 15 

"  — I,  Drawing  Plate. ..  15 

"         —II,  Drawing  Plate..  15 

"         and  section  lining 14 

Set  of  plans 15 

Shade  lines  13 

"      lines 15 

Shaft  flange  coupling 14 

Sharpening  the  drawing  pen.. .  13 

Shrinking  fit 14 

Side  elevation  ..   13 

"     elevation 15 

"     view 13 

"     view 15 

Sight,  Line  of  15 

Single  V-threaded  screw^ ..4 

Solids,  Classification  of 26 

"        Development  of 1& 

"        Development  of  inter- 
sected   16 

*        Development   of    intri- 
cate    16 

"        How    divided    into 

classes 16 

"        Irregular 16 

"        Irregular 16 

*'        Irregular 16 

"        Projection  of 15 

"  Projection  of  irregular.  15 
"  Revolution  of  radial...  16 
"  Sections  of  irregular. . .  15 
"  Stretchouts  for  parallel  16 
'•  Stretchouts  for  radial. .  16 
Sphere  and  cylinder.  Intersec- 
tion of 15 

"        and  cylinder.  Intersec- 
tion of 16 

"       Development  of 16 

"        Sections  of 15 

Spur  gear-wheels 14 

Square  cast-iron  washer 13 

'•       To  inscribe  in  circle. . .  13 

"        headed  bolt 13 

Straight  line  equal  in  length  to 

arc  of  circle   13 


73 

3 

2 

71 

77 

78 

75 

75 

85 

71 

9 

81 

70 

81 

2 

10 

73 

4 

23 

14 

10 

52 

9 

52 

9 

6 

17 

3 

5 


10 
54 
56 
57 
55 
78 
40 
78 
14 
39 

99 

35 

64 
71 
55 
58 
3S 
58 


Sec.  Page 
Straight  line  into  equal  parts, 

To  divide  a 13  &4 

"         line.  To  bisect  a 13  28 

"        line.  To  draw  a  paral- 
lel to  a 13  30 

"         line.  To  draw   a  per- 
pendicular to  a 13  29 

"        line.  To  draw'  a  per- 
pendicular to  a 13  30 

Stretchout 16  14 

"           for  parallel  solids. .  16  14 

"            for  radial  solids 16  39 

"           Laying  off 16  16 

Stringers 14  29 

Surfaces  bounded  by  curved 

lines 15  44 

Dark 13  74 

Hidden 15  58 

"          in  a  pattern.  How  re- 
lated   16  1 

"          Inclined  position  of..  15  40 
'■         Intersection   of    two 

cylindrical 13  68 

'■         Irregularly  outlined  15  48 

Light 13  74 

"         Oblique  irregular...  15  54 

"          Oblique,  Position  of.  15  49 

"          Plane   15  36 

"          Projectifin  of 15  48 

"          Projection  of 15  54 

"          Projection  of 15  49 

T  Sec.  Pag* 

T  joint.  Pattern  for 16  25 

'•square 13  1 

Teeth,  Cycloidal 14  51 

Third-angle  projection 15  7 

"      angle  projection,   Read- 
ing       15  57 

Threads,  Hidden  screw 14  8 

Timber  trestle 14  27 

Tool  finish.... 14  10 

"      Roughing  box 14  37 

Tooth,  Profile  of  a 14  53 

Top  plan 13  51 

"     view 13  51 

Tracing  cloth 14  77 

"        paper 14  77 

Tracings 14  77 

Trestle,  Compound 14  27 

Timber 14  27 

Triangle 15  81 

"         To  draw  an  equilateral  13  36 
"          To    draw    from    two 
sides   and    included 

angle 13  37 


INDEX 


XV 


Sec.  Pa^e 

Triangles 13         3 

Triangulation 16  49 

"  Basis  of 15  33 

"              Modified    meth- 
ods of 16  60 

True  edge  lines 16  41 

"      length      of     lines,     How 

found 15  35 

Turned,  Meaning  of 14  10 

Two-pieced  elbow,  Pattern  for  16  25 

V  Sec.  Page 

Vertex  of  parabola 13  45 

Vertical  plane 15         6 

"         plane 15         7 

View,  Side 13  51 

"      Top 13  51 


Sec.  Page 

Views,  Foreshortened 15        10 

Full 15       43 

"        necessary  in  projection  15         6 

"        Position  of  full 15        52 

W  Sec.  Page 

Washer,  Square  cast-iron 13       58 

Wedge 13        55 

Wheel,  Hand 14       20 

Working  drawing,  Reading  a..  14       66 

"         drawings 15  2 

"         drawings,  Kinds  of. .  14        10 

"         drawings.  To  read. ..  15       33 

Z  Sec.  Page 
Zones,  Development  of  sphere 

by IC       64 


